Sickle cell disease

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Sickle_cell_disease 

 

Sickle cell disease

Sickle cell disease (SCD), also simply called sickle cell, is a group of inherited hemoglobin-related blood disorders. The most common type is known as sickle cell anaemia. Sickle cell anaemia results in an abnormality in the oxygen-carrying protein haemoglobin found in red blood cells. This leads to the red blood cells adopting an abnormal sickle-like shape under certain circumstances. With this shape, they are unable to deform as they pass through capillaries, causing blockages.

Problems in sickle cell disease typically begin around 5 to 6 months of age. Several health problems may develop, such as attacks of pain (known as a sickle cell crisis) in joints, anaemia, swelling in the hands and feet, bacterial infections, dizziness and stroke. The probability of severe symptoms, including long-term pain, increases with age. Without treatment, people with sickle cell disease rarely reach adulthood, but with good healthcare, median life expectancy is between 58 and 66 years. All of the major organs are affected by sickle cell disease. The liver, heart, kidneys, lungs, gallbladder, eyes, bones, and joints can be damaged from the abnormal functions of the sickle cells and their inability to effectively flow through the small blood vessels.

Sickle cell disease occurs when a person inherits two abnormal copies of the β-globin gene that make haemoglobin, one from each parent. The abnormal gene generates haemoglobin S (HbS) which changes the properties of red blood cells. A sickle cell crisis occurs when red blood cells switch from the normal saucer-like shape to a sickle-like shape which can obstruct small blood vessels; an attack can be set off by temperature changes, stress, dehydration, and high altitude. A person with a single abnormal gene does not usually have symptoms and is said to have sickle cell trait, these people are also referred to as carriers. Diagnosis is by a blood test, and some countries test all babies at birth for the disease. Diagnosis of the unborn foetus is also possible during pregnancy.

The care of people with sickle cell disease may include infection prevention with vaccination and antibiotics, high fluid intake, folic acid supplementation, and pain medication. Other measures may include blood transfusion and the medication hydroxycarbamide (hydroxyurea). In 2023, new gene therapies were approved involving the genetic modification and replacement of blood forming stem cells in the bone marrow.

As of 2021, sickle cell disease is estimated to affect about 7.7 million people worldwide, directly causing an estimated 34,000 annual deaths and a contributory factor to a further 376,000 deaths. About 80% of sickle cell disease cases are believed to occur in sub-Saharan Africa. It also occurs to a lesser degree among people in parts of India, Southern Europe, West Asia, North Africa and among people of African origin (sub-Saharan) living in other parts of the world. The condition was first described in the medical literature by American physician James B. Herrick in 1910. In 1949, its genetic transmission was determined by E. A. Beet and J. V. Neel. In 1954, it was established that carriers of the abnormal gene are protected to some degree against malaria, which accounts for its persistence in populations threatened by malaria.

Signs and symptoms

Dactylitis in the hands of an infant

Sickle cells in human blood - both normal red blood cells and sickle-shaped cells are present.

Normal blood cells next to a sickle blood cell, coloured scanning electron microscope image

Signs of sickle cell disease usually begin in early childhood. The severity of symptoms can vary from person to person, as can the frequency of crisis events. Sickle cell disease may lead to various acute and chronic complications, several of which have a high mortality rate.

First events

When sickle cell disease presents within the first year of life, the most common problem is an episode of pain and swelling in the child's hands and feet, known as dactylitis or "hand-foot syndrome". Pallor, jaundice, and fatigue can also be early signs due to anaemia resulting from sickle cell disease.

In children older than 2 years, the most common initial presentation is a painful episode of a generalised or variable nature, while a slightly less common presentation involves acute chest pain. Dactylitis is rare or almost never occurs in children over the age of 2.

Critical events

Vaso-occlusive crisis

Also termed "sickle cell crisis" or "sickling crisis", the vaso-occlusive crisis (VOC) manifests principally as extreme pain, most often affecting the chest, back, legs, and/or arms. The underlying cause is sickle-shaped red blood cells that obstruct capillaries and restrict blood flow to an organ, resulting in ischaemia, pain, necrosis, and often organ damage. The frequency, severity, and duration of these crises vary considerably. Milder crises can be managed with nonsteroidal anti-inflammatory drugs. For more severe crises, patients may require inpatient management for intravenous opioids. Vaso-occlusive crisis involving organs such as the lungs or the penis are considered an emergency and treated with red blood cell transfusions.

A VOC can be triggered by anything which causes blood vessels to constrict; this includes physical or mental stress, cold, and dehydration. "After Haemoglobin S (HbS) deoxygenates in the capillaries, it takes some time (seconds) for HbS polymerization and the subsequent flexible-to-rigid transformation. If the transit time of RBC through the microvasculature is longer than the polymerization time, sickled RBC will lodge in the microvasculature."

Splenic sequestration crisis

The spleen is especially prone to damage in sickle cell disease due to its role as a blood filter. A splenic sequestration crisis, also known as a spleen crisis, is a medical emergency that occurs when sickled red blood cells block the spleen's filter mechanism, causing the spleen to swell and fill with blood. The accumulation of red blood cells in the spleen results in a sudden drop in circulating haemoglobin and potentially life-threatening anaemia. Symptoms include pain on the left side, swollen spleen (which can be detected by palpation), fatigue, dizziness, irritability, rapid heartbeat, or pale skin. It most commonly affects young children; the median age of first occurrence is 1.4 years. By the age of 5 years, repeated instances of sequestration cause scarring and eventual atrophy of the spleen.

Treatment is supportive, with blood transfusion if haemoglobin levels fall too low. Full or partial splenectomy may be necessary. Long term consequences of a loss of spleen function are increased susceptibility to bacterial infections.

Acute chest syndrome

Acute chest syndrome is caused by a VOC which affects the lungs, possibly triggered by infection or by emboli which have circulated from other organs. Symptoms include wheezing, chest pain, fever, pulmonary infiltrate (visible on x-ray), and hypoxemia. After sickling crisis (see above), it is the second-most common cause of hospitalisation, and it accounts for about 25% of deaths in patients with sickle cell disease. Most cases present with vaso-occlusive crises and then develop acute chest syndrome.

Aplastic crisis

Aplastic crises are instances of an acute worsening of the patient's baseline anaemia, producing pale appearance, fast heart rate, and fatigue. This crisis is normally triggered by parvovirus B19, which directly affects production of red blood cells by invading the red cell precursors and multiplying in and destroying them. Parvovirus infection  almost completely prevents red blood cell production for two to three days (red cell aplasia). In normal individuals, this is of little consequence, but the shortened red cell life of people with sickle cell disease results in an abrupt, life-threatening situation. Reticulocyte count drops dramatically during the disease (causing reticulocytopenia), red cell production lapses, and the rapid destruction of existing red cells leads to acute and severe anaemia. This crisis takes four to seven days to resolve. Most patients can be managed supportively; some need a blood transfusion.

Complications

Sickle cell anaemia can lead to various complications, including:

• An increased risk of severe bacterial infections is due to the loss of functioning spleen tissue. These infections are typically caused by bacteria such as Streptococcus pneumoniae and Haemophilus influenzae. Daily penicillin prophylaxis is the most commonly used treatment during childhood, with some haematologists continuing treatment indefinitely. Patients benefit from routine vaccination for S. pneumoniae.
• Stroke can result from blockage of blood vessels in the brain, causing numbness, confusion, or weakness, which may be long-lasting. Silent stroke causes no immediate symptoms, but is associated with damage to the brain. Silent stroke is probably five times as common as symptomatic stroke. About 10–15% of children with sickle cell disease have strokes, with silent strokes predominating in the younger patients.
• Cholelithiasis (gallstones) and cholecystitis may result from excessive bilirubin production and precipitation due to prolonged haemolysis.
• Avascular necrosis (aseptic bone necrosis) of the hip and other major joints may occur as a result of ischaemia.
• Priapism and infarction of the penis.
• Osteomyelitis (bacterial bone infection) as a result of damage to the spleen, commonly caused by either Staphylococcus aureus or species of Salmonella.
• Chronic kidney failure due to sickle-cell nephropathy manifests itself with hypertension, protein loss in the urine, loss of red blood cells in urine, and worsened anaemia. If it progresses to end-stage kidney failure, it carries a poor prognosis.
• Leg ulcers are relatively common in sickle cell disease and can be disabling.
• In the eyes, background retinopathy, proliferative retinopathy, vitreous haemorrhages, and retinal detachments can result in blindness.
• During pregnancy, intrauterine growth restriction, spontaneous abortion, and pre-eclampsia
• Chronic pain: Even in the absence of acute vaso-occlusive pain, many patients have unreported chronic pain.
• Pulmonary hypertension (increased pressure on the pulmonary artery) can lead to strain on the right ventricle and a risk of heart failure; typical symptoms are shortness of breath, decreased exercise tolerance, and episodes of syncope. Evidence of pulmonary hypertension is found in 21% of children and 30% of adults when tested; this is associated with reduced walking distance and increased mortality.
• Diastolic dysfunction in the left ventricle and cardiomyopathy, caused by fibrosis or scarring of cardiac tissues. This also contributes to pulmonary hypertension, decreased exercise capacity, and arrhythmias.

Genetics

See also: Introduction to genetics

See also: Price equation examples § Evolution of sickle cell disease

Autosomal recessive inheritance means acquiring two changed genes from each parent. If both parents are carriers for the autosomal recessive gene, there is a 25% chance of their child having and expressing the disorder. Other children will be unaffected, but may be carriers.

Base-pair substitution that causes sickle cell anaemia

Haemoglobin is an oxygen-binding protein, found in erythrocytes, which transports oxygen from the lungs (or in the foetus, from the placenta) to the tissues. Each molecule of haemoglobin comprises 4 protein subunits, referred to as globins. Normally, humans have:-

• haemoglobin F (foetal haemoglobin, HbF), consisting of two alpha (α-globin) and two gamma (γ-globin) chains. This dominates during the development of the foetus and until about 6 weeks of age. Afterwards, haemoglobin A dominates throughout life.
• haemoglobin A (adult haemoglobin, HbA) which consists of two alpha and two beta (β-globin) chains. This is the most common human haemoglobin tetramer, accounting for over 97% of the total red blood cell haemoglobin in normal adults.
• Haemoglobin B2 (HbA2) is a second form of adult haemoglobin and is composed of two alpha and two delta (δ-globin) chains. This haemoglobin typically makes up 1–3% of haemoglobin in adults.

β-globin is encoded by the HBB gene on human chromosome 11; mutations in this gene produce variants of the protein which are implicated with abnormal hemoglobins. The mutation that causes sickle cell disease results in an abnormal haemoglobin known as haemoglobin S (HbS), which replaces HbA in adults. The human genome contains a pair of genes for β-globin; in people with sickle cell disease, both genes are affected, and the erythropoietic cells in the bone marrow will only create HbS. In people with sickle cell trait, only one gene is abnormal; erythropoiesis generates a mixture of normal HbA and sickle HbS. The person has very few, if any, symptoms of sickle cell disease but carries the gene and can pass it on to their children.

Sickle cell disease has an autosomal recessive pattern of inheritance from parents. Both copies of the affected gene must carry the same mutation (homozygous condition) for a person to be affected by an autosomal recessive disorder. An affected person usually has unaffected parents who each carry one mutated gene and one normal gene (heterozygous condition) and are referred to as genetic carriers; they may not have any symptoms. When both parents have the sickle cell trait, any given child has a 25% chance of sickle cell disease; a 25% chance of no sickle cell alleles, and a 50% chance of the heterozygous condition (see diagram).

There are several different haplotypes of the sickle cell gene mutation, indicating that it may have arisen spontaneously in different geographic areas. The variants are known as Cameroon, Senegal, Benin, Bantu, and Saudi-Asian. These are clinically important because some are associated with higher HbF levels, e.g., Senegal and Saudi-Asian variants, and tend to have milder disease.

The gene defect is a single nucleotide mutation of the β-globin gene, which results in the amino acid glutamate being substituted by valine at position 6 of the β-globin chain. Haemoglobin S with this mutation is referred to as HbS, as opposed to the normal adult HbA. Under conditions of normal oxygen concentration, this causes no apparent effects on the structure of haemoglobin or its ability to transport oxygen around the body. However, the deoxy form of HbS has an exposed hydrophobic patch, which causes HbS molecules to join to form long, inflexible chains. Under conditions of low oxygen concentration in the bloodstream, such as exercise, stress, altitude, or dehydration, HbS polymerisation forms fibrous precipitates within the red blood cell. In people homozygous for the sickle cell mutation, the presence of long-chain polymers of HbS distort the shape of the red blood cell from a smooth, doughnut-like shape to the sickle shape, making it fragile and susceptible to blocking or breaking within capillaries.

In people heterozygous for HbS (carriers of sickle cell disease), the polymerisation problems are minor because the normal allele can produce half of the haemoglobin. Sickle cell carriers have symptoms only if they are deprived of oxygen (for example, at altitude) or while severely dehydrated.

Malaria

Historical distribution of the sickle cell trait

Historical distribution of malaria (no longer endemic in Europe)

Modern distribution of malaria

SCD is most prevalent in areas in which malaria has historically been endemic. The sickle cell trait provides a carrier with a survival advantage against malaria fatality over people with normal haemoglobin in regions where malaria is endemic.

Infection with the malaria parasite affects asymptomatic carriers of the abnormal haemoglobin gene differently from people with full sickle cell disease. Carriers (heterozygous for the gene) who catch malaria are less likely to suffer from severe symptoms than people with normal haemoglobin. People with sickle cell disease (homozygous for the gene) are similarly less likely to become infected with malaria; however, once infected, they are more likely to develop severe and life-threatening anaemia.

The impact of sickle cell anaemia on malaria immunity illustrates some evolutionary trade-offs that have occurred because of endemic malaria. Although the shorter life expectancy for those with the homozygous condition would tend to disfavour the trait's survival, the trait is preserved in malaria-prone regions because of the benefits provided by the heterozygous form; an example of natural selection.

Due to the adaptive advantage of the heterozygote, the disease is still prevalent, especially among people with recent ancestry in malaria-stricken areas, such as Africa, the Mediterranean, India, and the Middle East. Malaria was historically endemic to southern Europe, but it was declared eradicated in the mid-20th century, except rare sporadic cases.

The malaria parasite has a complex lifecycle and spends part of it in red blood cells. There are two mechanisms that protect sickle cell carriers from malaria. One is that the parasite is hindered from growing and reproducing in a carrier's red blood cells; another is that a carrier's red cells show signs of damage when infected, and are detected and destroyed as they pass through the spleen.

Pathophysiology

Scanning electron micrograph showing a mixture of red blood cells, some with round normal morphology, some with mild sickling showing elongation and bending

Under conditions of low oxygen concentration, haemoglobin S polymerises to form long strands within the red blood cell (RBC). These strands distort the shape of the cell and, after a few seconds, cause it to adopt an abnormal, inflexible, sickle-like shape. This process reverses when oxygen concentration is raised and the cells resume their normal biconcave disc shape. If sickling takes place in the venous system, after blood has passed through the capillaries, it does not affect the organs, and the RBCs can unsickle when they become oxygenated in the lungs. Repeated switching between sickle and normal shapes damages the membrane of the RBC so that it eventually becomes permanently sickled.

Normal red blood cells are quite elastic and have a biconcave disc shape, which allows the cells to deform to pass through capillaries. In sickle cell disease, low oxygen tension promotes red blood cell sickling and repeated episodes of sickling damage the cell membrane and decrease the cell's elasticity. These cells fail to return to normal shape when normal oxygen tension is restored. As a consequence, these rigid blood cells are unable to deform as they pass through narrow capillaries, leading to vessel occlusion and ischaemia.

Cells that have become sickled are detected as they pass through the spleen and are destroyed. In young children with sickle cell disease, the accumulation of sickled cells in the spleen can result in splenic sequestration crisis. In this, the spleen becomes engorged with blood, depriving the general circulation of blood cells and leading to severe anaemia. The spleen initially becomes noticeably swollen, but the lack of a healthy blood flow through the organ culminates in scarring of the spleen tissues and eventually death of the organ, generally before the age of 5 years.

The actual anaemia of the illness is caused by haemolysis, the destruction of the red cells, because of their shape. Although the bone marrow attempts to compensate by creating new red cells, it does not match the rate of destruction. Healthy red blood cells typically function for 90–120 days, but sickled cells only last 10–20 days.

The rapid breakdown of RBCs in sickle cell disease results in the release of free heme into the bloodstream, exceeding the capacity of the body's protective mechanisms. Although heme is an essential component of haemoglobin, it is also a potent oxidative molecule. Free heme is also an alarmin – a signal of tissue damage or infection, which triggers defensive responses in the body and increases the risk of inflammation and vaso-occlusive events.

Diagnosis

Prenatal and newborn screening

Checking for sickle cell disease begins during pregnancy, with a prenatal screening questionnaire which includes, among other things, a consideration of health issues in the child's parents and close relatives. During pregnancy, genetic testing can be done on either a blood sample from the foetus or a sample of amniotic fluid. During the first trimester of pregnancy, chorionic villus sampling (CVS) is also a technique used for sickle cell disease prenatal diagnosis. A routine heel prick test, in which a small sample of blood is collected a few days after birth, is used to check conclusively for sickle cell disease as well as other inherited conditions.

Tests

A schematic of haemoglobin electrophoresis, showing the banding which is typical of various types of haemoglobin. Note that sickle cell disease (SCD) gives a single, bold band whereas sickle cell trait gives two slightly fainter bands.

Where sickle cell disease is suspected, a number of tests can be used. Often, a simpler, cheaper test is applied first, with a more complex test, such as DNA analysis, used to confirm a positive result.

Two tests that are specific for sickle cell disease:

• A blood smear is a thin layer of blood smeared on a glass microscope slide and then stained in such a way as to allow the various blood cells to be examined microscopically. This technique can be used to visually detect sickled cells; however, it does not detect sickle cell carriers.
• A solubility test relies on the fact that HbS is less soluble than normal haemoglobin (HbA); it is highly reliable but does not distinguish between full sickle cell disease and carrier status.

Tests which can be used for sickle cell disease as well as for other hemoglobinopathies:

• Haemoglobin electrophoresis is a test that can detect different types of haemoglobin. Haemoglobin is extracted from the red cells, then introduced into a porous gel and subjected to an electrical field. This separates the normal and abnormal types of haemoglobin, which can then be identified and quantified.
• Isoelectric focusing (IEF) is a technique that can be used to diagnose sickle cell disease and other hemoglobinopathies. The technique separates molecules based on their isoelectric point, or the pH at which they have no net electrical charge. IEF uses an electric charge to separate and identify different types of haemoglobin, which become focused into sharp, stationary bands. The technique can distinguish many types of abnormal haemoglobin.
• High-performance liquid chromatography (HPLC) is reliable, fully automated, and able to distinguish most types of sickle cell disease, including heterozygous. The method separates and quantifies haemoglobin fractions by measuring their rate of flow through a column of absorbent material.
• DNA analysis using polymerase chain reaction (PCR), to amplify small samples of DNA. Variants of PCR used to diagnose sickle cell disease include amplification-refractory mutation system (ARMS) and allele-specific recombinase polymerase amplification. These tests can identify subtypes of sickle cell disease as well as combination hemoglobinopathies.

Genetic counselling

Genetic counselling is the process by which people with a hereditary disorder are advised of the probability of transmitting it and how this may be prevented or ameliorated.

People who are known carriers of the disease or at risk of having a child with sickle cell anaemia may undergo genetic counselling. Genetic counsellors work with families to discuss the benefits, limitations, and logistics of genetic testing options as well as the potential impact of testing and test results on the individual. Counselling is best given before a child is conceived, and several possible courses could be suggested. These include adoption, the use of eggs or sperm from a healthy donor, and in-vitro fertilisation (IVF) when combined with pre-implantation genetic diagnosis of the embryos.

Treatment

Further information: Pain management in children

Management

Several precautions can help reduce the risk of developing a sickling crisis. Lifestyle behaviours include maintaining good hydration and avoiding physical stress or exhaustion. Since sickling can be triggered by low oxygen levels, people with sickle cell disease should avoid high altitudes such as high mountains or flying in unpressurised aircraft. People with sickle cell disease should avoid alcohol and smoking, as alcohol can cause dehydration and smoking can trigger acute chest syndrome. Stress can also trigger a sickle cell crisis, so relaxation techniques like breathing exercises can help.

Pneumococcal infection is a leading cause of death among children with sickle cell disease; penicillin is recommended daily during the first 5 years of life to minimise the risk of infection.

Dietary supplementation of folic acid is sometimes recommended, on the basis that it facilitates the creation of new red blood cells and may reduce anaemia. A Cochrane review of its use in 2016 found "the effect of supplementation on anaemia and any symptoms of anaemia remains unclear" due to a lack of medical evidence.

People with sickle cell disease are recommended to receive all vaccinations which are recommended by health authorities to avoid serious infection, which might trigger a sickling crisis.

Hydroxyurea was the first approved drug for the treatment of sickle cell disease, which has been shown to decrease the number and severity of attacks and possibly increase survival time. This is achieved, in part, by reactivating foetal haemoglobin production in place of the haemoglobin S that causes sickling. Hydroxyurea lowers the expression of adhesion molecules on endothelial and red blood cells, which lowers the chance of small vessel blockages. Additionally, it encourages the release of nitric oxide, which enhances blood flow and inhibits the formation of clots. Hydroxyurea had previously been used as a chemotherapy agent, and some concern exists that long-term use may be harmful. A Cochrane review in 2022 found a weak evidence base for its use in sickle cell disease.

Voxelotor was received accelerated approval as a treatment for sickle cell disease in the United States in 2019, and was approved by the European Medicines Agency (EMA) in 2021. In trials, it had been shown to have disease-modifying potential by increasing haemoglobin levels and decreasing hemolysis indicators.  However, following an increased risk of vaso-occlusive seizures and death observed in registries and clinical trials, the manufacturer, Pfizer, withdrew it from the market worldwide.

Blood transfusion

Main article: Transfusion therapy (Sickle-cell disease)

The simple, or top-up transfusion, is a procedure in which healthy blood cells from a donor are infused into the patient's bloodstream. This benefits by alleviating anaemia and increasing oxygen levels in the tissues, reducing the risk of sickling and relieving sickling symptoms. A simple transfusion can be used to treat sickle cell disease when haemoglobin levels drop too low, or to prepare for an operation or pregnancy. It can also be used to protect against long-term complications or to reduce the risk of stroke.

An exchange transfusion is a procedure in which blood is removed from the body, then processed to extract sickled cells, which are replaced by healthy red blood cells from a donor. The treated blood, including white cells and plasma, is then returned to the patient. Exchange transfusions are likely to be needed in an emergency, in severe cases of sickle cell disease, or to support a mother during pregnancy.

Stroke prevention

Transcranial Doppler ultrasound (TCD) can detect children with sickle cell who have a high risk for stroke. The ultrasound test detects blood vessels partially obstructed by sickle cells by measuring the rate of blood into the brain, as blood flow velocity is inversely related to arterial diameter, and consequently, high blood flow velocity is correlated with narrowing of the arteries.

In children, preventive RBC transfusion therapy has been shown to reduce the risk of first stroke or silent stroke when transcranial Doppler ultrasonography shows abnormal cerebral blood flow. In those who have sustained a prior stroke event, it also reduces the risk of recurrent stroke and additional silent strokes.

Vaso-occlusive crisis

Most people with sickle cell disease have intensely painful episodes called vaso-occlusive crises (VOC). However, the frequency, severity, and duration of these crises vary tremendously. In a VOC, the circulation of blood vessels is obstructed by sickled red blood cells, causing ischemic injuries to the tissues, inflammation, and pain. Recurrent episodes may cause irreversible organ damage.

The most common and obvious symptom of a VOC is pain, which may be felt anywhere in the body but most commonly in the limbs and back. The degree of pain varies from mild to severe. Home treatment options include bed rest and hydration, and pain control using over-the-counter medication such as paracetamol or ibuprofen. More severe cases may require prescription opioids such as codeine or morphine for pain control.

In 2019, crizanlizumab, a monoclonal antibody targeting P-selectin, was approved in the United States to reduce the frequency of vaso-occlusive crisis in those 16 years and older. It had also been approved in the UK and the European Union, but in both cases authorisation was subsequently withdrawn because of poor evidence of its effectiveness.

Acute chest syndrome

Acute chest syndrome is caused by vaso-occlusion occurring in the lungs. As with a VOC, treatment includes pain control and hydration. Antibiotics are required because there is a severe risk of pulmonary infection, and oxygen supplementation for hypoxia. Blood transfusion may also be required, or exchange transfusion in severe cases.

Treating avascular necrosis

When treating avascular necrosis of the bone in people with sickle cell disease, treatment aims to reduce or stop the pain and maintain joint mobility. Treatment options include resting the joint, physical therapy, pain-relief medicine, joint-replacement surgery, or bone grafting.

Psychological therapy

Psychological therapies such as patient education, cognitive therapy, behavioural therapy, and psychodynamic psychotherapy, that aim to complement current medical treatments, require further research to determine their effectiveness.

Stem cell treatments

Hematopoietic stem cells (HSC) are cells in the bone marrow that can develop into all types of blood cells, including red blood cells, white blood cells, and platelets. There are two possible ways to treat sickle cell disease and some other hemoglobinopathies by targeting HSCs. Since 1991, a small number of patients have received bone marrow transplants from healthy matched donors, although this procedure has a high level of risk. More recently, it has become possible to use CRISPR gene editing technology to modify the patient's own HSCs in a way that reduces or eliminates the production of sickle haemoglobin HbS and replaces it with a non-sickling form of haemoglobin.

All stem cell treatments must involve myeloablation of the patients' bone marrow to remove HSCs containing the faulty gene. This requires high doses of chemotherapy agents with side effects such as sickness and tiredness. A long hospital stay is necessary after infusion of the replacement HSCs, while the cells take up residence in the bone marrow and start to make red blood cells with the stable form of haemoglobin.

Gene therapy

Gene therapy was first trialled in 2014 on a single patient, and followed by clinical trials in which several patients were successfully treated. In 2023, both exagamglogene autotemcel (Casgevy) and lovotibeglogene autotemcel (Lyfgenia) were approved for the treatment of sickle cell disease. Kendric Cromer in October 2024 became the first commercial case in the US to receive gene therapy and was discharged from Children's National Hospital. The one-off gene-editing therapy, Casgevy, also known as Exa-cel, is to be offered to patients on the National Health Service (NHS) in England as from 2025.

Both Casgevy and Lyfgenia work by first harvesting the patient's HSCs, then using CRISPR gene editing to modify their DNA in the laboratory. In parallel with this, the person with sickle cell disease's bone marrow is put through a myeloablation procedure to destroy the remaining HSCs. The treated cells are then infused back into the patient, where they colonise the bone marrow and eventually resume production of blood cells. Casgevy works by editing the BCL11A gene, which normally inhibits the production of haemoglobin F (foetal haemoglobin) in adults. The edit has the effect of increasing the production of HbF, which is not prone to sickling. Lyfgenia introduces a new gene for T87Q-globin, which coexists with the sickling beta-globin but reduces the incidence of sickling.

Hematopoietic stem cell transplantation

Hematopoietic stem cell transplantation (HSCT) involves replacing the dysfunctional stem cells from a person with sickle cell disease with healthy cells from a well-matched donor. Finding a well matched donor is essential to the process' success. Different types of donors may be suitable and include umbilical cord blood, human leukocyte antigen (HLA) matched relatives, or HLA matched donors that are not related to the person being treated. Risks associated with HSCT can include graft-versus-host disease, failure of the graft, and other toxicity related to the transplant.

Prognosis

Sickle cell disease is most prevalent in sub-saharan Africa. In areas without healthcare infrastructure, it is estimated that between 50% and 90% of children born with the disease die before the age of 5 years.

In contrast, life expectancy in the United States in 2010–2020 was 43 years and in the UK 67 years.

Epidemiology

The HbS gene can be found in every ethnic group. The highest frequency of sickle cell disease is found in tropical regions, particularly sub-Saharan Africa, tribal regions of India, and the Middle East. About 80% of sickle cell disease cases are believed to occur in Sub-Saharan Africa. Migration of substantial populations from these high-prevalence areas to low-prevalence countries in Europe has dramatically increased in recent decades and in some European countries, sickle cell disease has now overtaken more familiar genetic conditions such as haemophilia and cystic fibrosis. In 2015, it resulted in about 114,800 deaths.

Sickle cell disease occurs more commonly among people whose ancestors lived in tropical and subtropical sub-Saharan regions where malaria is or was common. Where malaria is common, carrying a single sickle cell allele (trait) confers a heterozygote advantage; humans with one of the two alleles of sickle cell disease show less severe symptoms when infected with malaria.

This condition is inherited in an autosomal recessive pattern, which means both copies of the gene in each cell have mutations. The parents each carry one copy of the mutated gene, but they typically do not show signs and symptoms of the condition.

Africa

Three-quarters of sickle cell cases occur in Africa. A World Health Organization report dated 2006 estimated that around 2% of newborns in Nigeria are affected by sickle cell anaemia, giving a total of 150,000 affected children born every year in Nigeria alone. The carrier frequency ranges between 10 and 40% across equatorial Africa, decreasing to 1–2% on the North African coast and <1% in South Africa. In the West African countries of Ghana and Nigeria, the frequencies can vary from 15 to 30%. In Nigeria, 24% of the population carries the gene, and 20 per 1,000 newborns are born with the disease, or 150,000 annually.

Uganda has the fifth-highest sickle cell disease burden in Africa. One study indicates that 20,000 babies per year, or 0.7% of the total, are born with sickle cell disease, and 13.3% carry the trait. In Uganda, carrier frequency of the trait varies strongly across tribal lines: among the Baamba, it reaches 45%.

United States

The number of people with the disease in the United States is about 100,000 (one in 3,300), mostly affecting Americans of sub-Saharan African descent. In the United States, about one out of 365 African-American children and one in every 16,300 Hispanic-American children have sickle cell anaemia. The life expectancy for men with sickle cell disease is approximately 42 years of age while women live approximately six years longer. An additional 2 million are carriers of the sickle cell trait. Most infants with sickle cell disease born in the United States are identified by routine neonatal screening. As of 2016, all 50 states include screening for sickle cell disease as part of their newborn screening. The newborn's blood is sampled through a heel-prick and is sent to a lab for testing. The baby must have been eating for a minimum of 24 hours before the heel-prick test can be done. Some states also require a second blood test to be done when the baby is two weeks old to ensure the results.

Sickle cell disease is the most common genetic disorder among African Americans. Approximately 8% are carriers and 1 in 375 are born with the disease. Patient advocates for sickle cell disease have complained that it gets less government and private research funding than similar rare diseases such as cystic fibrosis, with researcher Elliott Vichinsky saying this shows racial discrimination or the role of wealth in health care advocacy. Overall, without considering race, approximately 1.5% of infants born in the United States are carriers of at least one copy of the mutant (disease-causing) gene.

France

Percentage of newborns screened for sickle cell disease within mainland France from 2006 to 2018

As a result of population growth in African-Caribbean regions of overseas France and immigration from North and sub-Saharan Africa to mainland France, sickle cell disease has become a major health problem in France. Sickle cell disease has become the most common genetic disease in the country, with an overall birth prevalence of one in 2,415 in mainland France, ahead of phenylketonuria (one in 10,862), congenital hypothyroidism (one in 3,132), congenital adrenal hyperplasia (one in 19,008) and cystic fibrosis (one in 5,014) for the same reference period.

Percentage of newborns screened regionally and overall for sickle cell disease in mainland France and fraction of positive outcomes in all screened in 2018

Since 2000, neonatal screening of sickle cell disease has been performed at the national level for all newborns defined as being "at-risk" for sickle cell disease based on ethnic origin (defined as those born to parents originating from sub-Saharan Africa, North Africa, the Mediterranean area (South Italy, Greece, and Turkey), the Arabic peninsula, the French overseas islands, and the Indian subcontinent). Since 3 August 2024, this screening is systematically applied to all newborns in France.

United Kingdom

In the United Kingdom, between 12,000 and 15,000 people are thought to have sickle cell disease  with an estimated 250,000 carriers of the condition in England alone. As the number of carriers is only estimated, all newborn babies in the UK receive a routine blood test to screen for the condition. Due to many adults in high-risk groups not knowing if they are carriers, pregnant women and both partners in a couple are offered screening so they can get counselling if they have the sickle cell trait. In addition, blood donors from those in high-risk groups are also screened to confirm whether they are carriers and whether their blood filters properly. Donors who are found to be carriers are informed and their blood, while often used for those of the same ethnic group, is not used for those with sickle cell disease who require a blood transfusion.

West Asia

In Saudi Arabia, about 4.2% of the population carries the sickle cell trait, and 0.26% have sickle cell disease. The highest prevalence is in the Eastern province, where approximately 17% of the population carries the gene and 1.2% have sickle cell disease. In 2005, Saudi Arabia introduced a mandatory premarital test including HB electrophoresis, which aimed to decrease the incidence of sickle cell disease and thalassemia.

In Bahrain, a study published in 1998 that covered about 56,000 people in hospitals in Bahrain found that 2% of newborns have sickle cell disease, 18% of the surveyed people have the sickle cell trait, and 24% were carriers of the gene mutation causing the disease. The country began screening of all pregnant women in 1992, and newborns started being tested if the mother was a carrier. In 2004, a law was passed requiring couples planning to marry to undergo free premarital counselling. These programmes were accompanied by public education campaigns.

India and Nepal

Sickle cell disease is common in some ethnic groups of central India, where the prevalence has ranged from 9.4 to 22.2% in endemic areas of Madhya Pradesh, Rajasthan, and Chhattisgarh. It is also endemic among Tharu people of Nepal and India; however, they have a sevenfold lower rate of malaria despite living in a malaria infested zone.

Caribbean Islands

In Jamaica, 10% of the population carries the sickle cell gene, making it the most prevalent genetic disorder in the country.

History

The first modern report of sickle cell disease may have been in 1846, where the autopsy of an executed runaway slave was discussed; the key finding was the absence of the spleen. Reportedly, African slaves in the United States exhibited resistance to malaria, but were prone to leg ulcers. The abnormal characteristics of the red blood cells, which later lent their name to the condition, was first described by Ernest E. Irons (1877–1959), intern to Chicago cardiologist and professor of medicine James B. Herrick (1861–1954), in 1910. Irons saw "peculiar elongated and sickle-shaped" cells in the blood of a man named Walter Clement Noel, a 20-year-old first-year dental student from Grenada. Noel had been admitted to the Chicago Presbyterian Hospital in December 1904 with anaemia. Noel was readmitted several times over the next three years for "muscular rheumatism" and "bilious attacks" but completed his studies and returned to the capital of Grenada (St. George's) to practice dentistry. He died of pneumonia in 1916 and is buried in the Catholic cemetery at Sauteurs in the north of Grenada. Shortly after the report by Herrick, another case appeared in the Virginia Medical Semi-Monthly with the same title, "Peculiar Elongated and Sickle-Shaped Red Blood Corpuscles in a Case of Severe Anemia." This article is based on a patient admitted to the University of Virginia Hospital on 15 November 1910. In the later description by Verne Mason in 1922, the name "sickle cell anemia" is first used. Childhood problems related to sickle cells disease were not reported until the 1930s, despite the fact that this cannot have been uncommon in African-American populations.

Memphis physician Lemuel Diggs, a prolific researcher into sickle cell disease, first introduced the distinction between sickle cell disease and trait in 1933, although until 1949, the genetic characteristics had not been elucidated by James V. Neel and E.A. Beet. 1949 was the year when Linus Pauling described the unusual chemical behaviour of haemoglobin S, and attributed this to an abnormality in the molecule itself. The molecular change in HbS was described in 1956 by Vernon Ingram. The late 1940s and early 1950s saw further understanding in the link between malaria and sickle cell disease. In 1954, the introduction of haemoglobin electrophoresis allowed the discovery of particular subtypes, such as HbSC disease.

Large-scale natural history studies and further intervention studies were introduced in the 1970s and 1980s, leading to widespread use of prophylaxis against pneumococcal infections among other interventions. Bill Cosby's Emmy-winning 1972 TV movie, To All My Friends on Shore, depicted the story of the parents of a child with sickle cell disease. The 1990s had the development of hydroxycarbamide, and reports of cure through bone marrow transplantation appeared in 2007.

Some texts refer to it as drepanocytosis.

Society and culture

United States

In the US, there can be stigma that hinders people with sickle cell disease from receiving necessary care; one element of this is attributed to racism as the majority of people with sickle cell disease are black. Due to this, in 1970, the Black Panther Party (also known as the Black Panther Party for Self Defense) opened dozens of free clinics across the U.S where free Sickle Cell Disease tests where offered. Over the course of the 1970s, thousands of people, largely African-Americans, received testing at one of these clinics.

In September 2017, the US Social Security Administration issued a policy interpretation ruling providing background information on sickle cell disease and a description of how Social Security evaluates the disease during its adjudication process for disability claims.

Uganda

Uganda has the fifth highest sickle cell disease (SCD) burden in the world. In Uganda, social stigma exists for those with sickle cell disease because of the lack of general knowledge of the disease. The general gap in knowledge surrounding sickle cell disease is noted among adolescents and young adults due to the culturally sanctioned secrecy about the disease. While most people have heard generally about the disease, a large portion of the population is relatively misinformed about how sickle cell disease is diagnosed or inherited. Those who are informed about the disease learned about it from family or friends and not from health professionals. Failure to provide the public with information about sickle cell disease results in a population with a poor understanding of the causes of the disease, symptoms, and prevention techniques. The differences, physically and socially, that arise in those with sickle cell disease, such as jaundice, stunted physical growth, and delayed sexual maturity, can also lead them to become targets of bullying, rejection, and stigma.

Rate of sickle cell disease in Uganda

The data compiled on sickle cell disease in Uganda has not been updated since the early 1970s. The deficiency of data is due to a lack of government research funds, even though Ugandans die daily from sickle cell disease. Data shows that the trait frequency of sickle cell disease is 20% of the population in Uganda. It is also estimated that about 25,000 Ugandans are born each year with sickle cell disease and 80% of those people do not live past five years old. Sickle cell disease also contributes 25% to the child mortality rate in Uganda. The Bamba people of Uganda, located in the southwest of the country, carry 45% of the gene which is the highest trait frequency recorded in the world. The Sickle Cell Clinic in Mulago is only one sickle cell disease clinic in the country and, on average, sees 200 patients a day.

Misconceptions about sickle cell disease

The stigma around the disease is particularly bad in regions of the country that are not as affected. For example, Eastern Ugandans tend to be more knowledgeable of the disease than Western Ugandans, who are more likely to believe that sickle cell disease resulted as a punishment from God or witchcraft. Other misconceptions about sickle cell disease include the belief that it is caused by environmental factors but, in reality, sickle cell disease is a genetic disease. There have been efforts throughout Uganda to address the social misconceptions about the disease. In 2013, the Uganda Sickle Cell Rescue Foundation was established to spread awareness of sickle cell disease and combat the social stigma attached to the disease. In addition to this organisation's efforts, there is a need for the inclusion of sickle cell disease education in preexisting community health education programmes in order to reduce the stigmatisation of sickle cell disease in Uganda.

Social isolation of people with sickle cell disease

The deeply rooted stigma of sickle cell disease in society causes families to often hide their family members' sick status for fear of being labeled, cursed, or left out of social events. Sometimes in Uganda, when it is confirmed that a family member has sickle cell disease, intimate relationships with all members of the family are avoided. The stigmatisation and social isolation that people with sickle cell disease tend to experience are often the consequence of popular misconceptions that people with sickle cell disease should not socialise with those free from the disease. This mentality robs people with sickle cell disease of the right to freely participate in community activities like everyone else. SCD-related stigma and social isolation in schools, especially, can make life for young people living with sickle cell disease challenging. For school-aged children living with sickle cell disease, the stigma they face can lead to peer rejection. Peer rejection involves the exclusion from social groups or gatherings. It often leads the excluded individual to experience emotional distress and may result in their academic underperformance, avoidance of school, and occupational failure later in life. This social isolation is also likely to negatively impact people with sickle cell disease's self-esteem and overall quality of life.

Mothers of children with sickle cell disease tend to receive disproportionate amounts of stigma from their peers and family members. These women will often be blamed for their child's diagnosis of sickle cell disease, especially if sickle cell disease is not present in earlier generations, due to the suspicion that the child's poor health may have been caused by the mother's failure to implement preventative health measures or promote a healthy environment for her child to thrive. The reliance on theories related to environmental factors to place blame on the mother reflects many Ugandans' poor knowledge of how the disease is acquired as it is determined by genetics, not environment. Mothers of children with sickle cell disease are also often left with very limited resources to safeguard their futures against the stigma of having sickle cell disease. This lack of access to resources results from their subordinating roles within familial structures as well as the class disparities that hinder many mothers' ability to satisfy additional childcare costs and responsibilities.

Women living with sickle cell disease who become pregnant often face extreme discrimination and discouragement in Uganda. These women are frequently branded by their peers as irresponsible for having a baby while living with sickle cell disease or even engaging in sex while living with sickle cell disease. The criticism and judgement these women receive, not only from healthcare professionals but also from their families, often leaves them feeling alone, depressed, anxious, ashamed, and with very little social support. Most pregnant women with sickle cell disease also go on to be single mothers as it is common for them to be left by their male partners who claim they were unaware of their partner's sickle cell disease status. Not only does the abandonment experienced by these women cause emotional distress for them, but this low level of parental support can be linked to depressive symptoms and overall lower quality of life for the child once they are born.

United Kingdom

The National Health Service (NHS) makes a number of treatments available for people with sickle cell disease. These include pain management, antibiotics and vaccinations to reduce the risk of infection, and blood transfusions where appropriate. Other treatments include Voxelotor which reduces the need for blood transfusions, and the gene therapy Casgevy. In England, a number of regional centres coordinate treatment for sickle cell disease and other haemoglobinopathies.

Media and arts representation of sickle cell disease

Popular media and art have been important educational tools about sickle cell disease. Hertz Nazaire was a Haitian-American visual artist, writer, and advocate who lived with sickle cell disease and used his art to raise awareness, combat stigma, and champion better access to care.

Representations of sickle cell disease in television shows include in the longrunning American medical drama ER (season 4 "Obstruction of Justice" and season 15 "Separation Anxiety" and "Dream Runner"), the 2024 British superhero series Supacell, in the 2024 period drama Lady in the Lake, and in the 2025 medical procedural drama The Pitt (episode "8:00 am"). Noah Wyle, who plays Dr. Michael "Robby" Robinavitch on The Pitt, has spoken about the importance of popular media representation of sickle cell and the impacts of racism on the quality of care patients receive.

Sickle cell disease has also been featured in several movies including the 1973 Sydney Poitier classic, A Warm December, 1996 Nigerian drama Mortal Inheritance, Genie Award- winning 2008 Canadian drama Nurse.Fighter.Boy, the 2017 animated documentary Spilled Milk, the 2020 Nigerian drama Strain, the 2022 animated short The Park Bench, and the 2023 American sci-fi action film Transformers: Rise of the Beasts.

Molecular symmetry

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Molecular_symmetry 

Symmetry elements of formaldehyde. C₂ is a two-fold rotation axis. σ_v and σ_v' are two non-equivalent reflection planes.

In chemistry, molecular symmetry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule's chemical properties, such as whether or not it has a dipole moment, as well as its allowed spectroscopic transitions. To do this it is necessary to use group theory. This involves classifying the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Symmetry is useful in the study of molecular orbitals, with applications to the Hückel method, to ligand field theory, and to the Woodward–Hoffmann rules. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry discuss symmetry. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.

There are many techniques for determining the symmetry of a given molecule, including X-ray crystallography and various forms of spectroscopy. Spectroscopic notation is based on symmetry considerations.

Point group symmetry concepts

Examples of the relationship between chirality and symmetry

Rotational axis (Cn)	Improper rotational elements (Sn)
	Chiral no Sn	Achiral mirror plane S1 = σ	Achiral inversion centre S2 = i
C1
C2

Elements

The point group symmetry of a molecule is defined by the presence or absence of 5 types of symmetry element.

• Symmetry axis: an axis around which a rotation by ${\displaystyle \frac{{360}^{\circ }}{n}}$ results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated C_n. Examples are the C₂ axis in water and the C₃ axis in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is aligned with the z-axis in a Cartesian coordinate system.
• Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is generated. This is also called a mirror plane and abbreviated σ (sigma = Greek "s", from the German 'Spiegel' meaning mirror). Water has two of them: one in the plane of the molecule itself and one equidistant from the two hydrogen atoms. A symmetry plane parallel with the principal axis is dubbed vertical (σ_v) and one perpendicular to it horizontal (σ_h). A third type of symmetry plane exists: If a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σ_d). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz).
• Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. In other words, a molecule has a center of symmetry when the points (x,y,z) and (−x,−y,−z) of the molecule always look identical. For example, whenever there is an oxygen atom in some point (x,y,z), then there also has to be an oxygen atom in the point (−x,−y,−z). There may or may not be an atom at the inversion center itself. An inversion center is a special case of having a rotation-reflection axis about an angle of 180° through the center. Examples are xenon tetrafluoride (a square planar molecule), where the inversion center is at the Xe atom, and benzene (C
  ₆H
  ₆) where the inversion center is at the center of the ring.
• Rotation-reflection axis: an axis around which a rotation by ${\displaystyle \frac{{360}^{\circ }}{n}}$, followed by a reflection in the plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated S_n. Examples are present in tetrahedral silicon tetrafluoride, with three S₄ axes, and the staggered conformation of ethane with one S₆ axis. An S₁ axis corresponds to a mirror plane σ and an S₂ axis is an inversion center i. A molecule which has no S_n axis for any value of n is a chiral molecule.
• Identity, abbreviated to E, from the German 'Einheit' meaning unity. This symmetry element simply consists of no change: every molecule has this symmetry element, which is equivalent to a C₁ proper rotation. It must be included in the list of symmetry elements so that they form a mathematical group, whose definition requires inclusion of the identity element. It is so called because it is analogous to multiplying by one (unity).

Point groups and their symmetry operationserrata - the 2-fold rotational axis C₂ in the dihedral groups D_n is horizontal not vertical (its usually taken to be the y axis)

Operations

The five symmetry elements have associated with them five types of symmetry operation, which leave the geometry of the molecule indistinguishable from the starting geometry. They are sometimes distinguished from symmetry elements by a caret or circumflex. Thus, Ĉ_n is the rotation of a molecule around an axis and Ê is the identity operation. A symmetry element can have more than one symmetry operation associated with it. For example, the C₄ axis of the square xenon tetrafluoride (XeF₄) molecule is associated with two Ĉ₄ rotations in opposite directions (90° and 270°), a Ĉ₂ rotation (180°) and Ĉ₁ (0° or 360°). Because Ĉ₁ is equivalent to Ê, Ŝ₁ to σ and Ŝ₂ to î, all symmetry operations can be classified as either proper or improper rotations.

For linear molecules, either clockwise or counterclockwise rotation about the molecular axis by any angle Φ is a symmetry operation.

Rotoreflection of an unspecified molecule with octahedral (left) and tetrahedral (right) symmetry

XeF₄, with square planar geometry, has 1 C₄ axis and 4 C₂ axes orthogonal to C₄. These five axes plus the mirror plane perpendicular to the C₄ axis define the D_4h symmetry group of the molecule.

Symmetry groups

Groups

The symmetry operations of a molecule (or other object) form a group. In mathematics, a group is a set with a binary operation that satisfies the four properties listed below.

In a symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of a C₄ rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C₄. By convention the order of operations is from right to left.

A symmetry group obeys the defining properties of any group.

1. closure property:
   For every pair of elements x and y in G, the product x*y is also in G.
   ( in symbols, for every two elements x, y ∈ G, x*y is also in G ).
   This means that the group is closed so that combining two elements produces no new elements. Symmetry operations have this property because a sequence of two operations will produce a third state indistinguishable from the second and therefore from the first, so that the net effect on the molecule is still a symmetry operation. This may be illustrated by means of a table. For example, the point group C₃ contains three symmetry operations: rotation by 120°, C₃, rotation by 240°, C₃² and rotation by 360°, which is equivalent to identity, E. The group C₃ is therefore not the same as the operation C₃, although the same notation is used.

   

   C_2v point group multiplication table

   Point group C₃ Multiplication table

   	E	C3	C32
   E	E	C3	C32
   C3	C3	C32	E
   C32	C32	E	C3

This table also illustrates the following properties
2. Associative property:
   For every x and y and z in G, both (x*y)*z and x*(y*z) result with the same element in G.
   (in symbols, (x*y)*z = x*(y*z) for every x, y, and z ∈ G)
3. existence of identity property:
   There must be an element (say e) in G such that product any element of G with e make no change to the element.
   (in symbols, x*e = e*x = x for every x ∈ G)
4. existence of inverse element:
   For each element x in G, there must be an element y in G such that product of x and y is the identity element e.
   (in symbols, for each x ∈ G there is a y ∈ G such that x*y = y*x = e for every x ∈ G)

The order of a group is the number of elements in the group. For groups of small orders, the group properties can be easily verified by considering its composition table, a table whose rows and columns correspond to elements of the group and whose entries correspond to their products.

Point groups

Flowchart for determining the point group of a molecule

The successive application (or composition) of one or more symmetry operations of a molecule has an effect equivalent to that of some single symmetry operation of the molecule. For example, a C₂ rotation followed by a σ_v reflection is seen to be a σ_v' symmetry operation: σ_v*C₂ = σ_v'. ("Operation A followed by B to form C" is written BA = C). Moreover, the set of all symmetry operations (including this composition operation) obeys all the properties of a group, given above. So (S,*) is a group, where S is the set of all symmetry operations of some molecule, and * denotes the composition (repeated application) of symmetry operations.

This group is called the point group of that molecule, because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarises all symmetry operations that all molecules in that category have. The symmetry of a crystal, by contrast, is described by a space group of symmetry operations, which includes translations in space.

Examples of point groups

Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl₃, POF₃, XeO₃, and NH₃ all share identical symmetry operations. They all can undergo the identity operation E, two different C₃ rotation operations, and three different σ_v plane reflections without altering their identities, so they are placed in one point group, C_3v, with order 6. Similarly, water (H₂O) and hydrogen sulfide (H₂S) also share identical symmetry operations. They both undergo the identity operation E, one C₂ rotation, and two σ_v reflections without altering their identities, so they are both placed in one point group, C_2v, with order 4. This classification system helps scientists to study molecules more efficiently, since chemically related molecules in the same point group tend to exhibit similar bonding schemes, molecular bonding diagrams, and spectroscopic properties.

Molecular image gallery

The following table shows a large number of molecules belonging to point groups labelled using Schoenflies notation. Each row contains example molecules a group belonging to the Schoenflies symbol on the left of the table In each row, the descriptions and examples have no higher symmetries, meaning that the named point group captures all of the point symmetries and is the highest order group applicable to that molecule. This table is excellent for an overall view of molecular forms but greater detail and the ability to move images is provided by the Otterbein site.

Point group	Symmetry operations	Simple description of typical geometry	Example 1	Example 2	Example 3
C1	E	no symmetry, chiral	bromochlorofluoromethane (both enantiomers shown)	lysergic acid	L-leucine and most other α-amino acids except glycine
Cs	E σ	mirror plane	thionyl chloride	hypochlorous acid	chloroiodomethane
Ci	E i	inversion center	meso-tartaric acid	mucic acid (meso-galactaric acid)	1,2 dibromo 1,2 dichloroethane
C∞v	E 2C∞Φ ∞σv	linear	hydrogen fluoride (and all other heteronuclear diatomic molecules)	nitrous oxide (dinitrogen monoxide)	hydrocyanic acid (hydrogen cyanide)
D∞h	E 2C∞Φ ∞σi i 2S∞Φ ∞C2	linear with inversion center	oxygen (and all other homonuclear diatomic molecules)	carbon dioxide	acetylene (ethyne)
C2	E C2	"open book geometry", chiral	hydrogen peroxide	hydrazine	tetrahydrofuran (twist conformation)
C3	E C3 C32	propeller, chiral	triphenylphosphine	triethylamine	phosphoric acid
C2h	E C2 i σh	planar with inversion center, no vertical plane	trans-1,2-dichloroethylene	trans-dinitrogen difluoride	trans-azobenzene
C2v	E C2 σv(xz) σv'(yz)	angular (H2O) or see-saw (SF4)	water	sulfur tetrafluoride	Dichloromethane
C3h	E C3 C32 σh S3 S35	propeller	boric acid	phloroglucinol (1,3,5-trihydroxybenzene)	benzotrifuroxan
C3v	E 2C3 3σv	trigonal pyramidal	ammonia (if pyramidal inversion is neglected)	phosphorus oxychloride	cobalt tetracarbonyl hydride, HCo(CO)4
C4v	E 2C4 C2 2σv 2σd	square pyramidal	xenon oxytetrafluoride	pentaborane(9), B5H9	nitroprusside anion [Fe(CN)5(NO)]2−
C5	E 2C5 2C52	five-fold rotational symmetry	C-reactive protein	[Fe(Me5-Cp)(P5)]	Corannulene derivative
C5v	E 2C5 2C52 5σv	'milking stool' complex	Cyclopentadienyl nickel nitrosyl (CpNiNO)	corannulene	Pentamethylcyclopentadienyl nickel nitrosyl (Cp*NiNO)
D2	E C2(x) C2(y) C2(z)	twist, chiral	biphenyl (skew conformation)	twistane (C10H16)	(δ,δ)-trans-[Co(en)2Cl2]+
D3	E C3(z) 3C2	triple helix, chiral	Tris(ethylenediamine)cobalt(III) cation	tris(oxalato)iron(III) anion	tris(en)cobalt(III)
D2h	E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)	planar with inversion center, vertical plane	ethylene	pyrazine	diborane
D3h	E 2C3 3C2 σh 2S3 3σv	trigonal planar or trigonal bipyramidal	boron trifluoride	phosphorus pentachloride	cyclopropane
D4h	E 2C4 C2 2C2' 2C2" i 2S4 σh 2σv 2σd	square planar	xenon tetrafluoride	octachlorodimolybdate(II) anion	Trans-[CoIII(NH3)4Cl2]+ (excluding H atoms)
D5h	E 2C5 2C52 5C2 σh 2S5 2S53 5σv	pentagonal	cyclopentadienyl anion	ruthenocene	C70
D6h	E 2C6 2C3 C2 3C2' 3C2‘’ i 2S3 2S6 σh 3σd 3σv	hexagonal	benzene	bis(benzene)chromium	coronene (C24H12)
D7h	E C7 S7 7C2 σh 7σv	heptagonal	tropylium (C7H7+) cation
D8h	E C8 C4 C2 S8 i 8C2 σh 4σv 4σd	octagonal	cyclooctatetraenide (C8H82−) anion	uranocene	bis(cot)thorium(IV)
D2d	E 2S4 C2 2C2' 2σd	90° twist	allene	tetrasulfur tetranitride	diborane(4) (excited state)
D3d	E 2C3 3C2 i 2S6 3σd	60° twist	ethane (staggered rotamer)	dicobalt octacarbonyl (non-bridged isomer)	cyclohexane chair conformation
D4d	E 2S8 2C4 2S83 C2 4C2' 4σd	45° twist	sulfur (crown conformation of S8)	dimanganese decacarbonyl (staggered rotamer)	octafluoroxenate ion (idealized geometry)
D5d	E 2C5 2C52 5C2 i 2S103 2S10 5σd	36° twist	ferrocene (staggered rotamer)	Ruthenocene (stag)	fulleride ion
S4	E 2S4 C2		1,2,3,4-tetrafluorospiropentane (meso isomer)	tetramethyl-cot	bis(dth)copper(I)
Td	E 8C3 3C2 6S4 6σd	tetrahedral	methane	phosphorus pentoxide	adamantane
Th	E 4C3 4C32 i 3C2 4S6 4S65 3σh	pyritohedron	[Fe(C5H5N)6]2+	[Th(NO3)6]2-	[Fe(H2O)6]2+/3+
Oh	E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh 6σd	octahedral or cubic	sulfur hexafluoride	molybdenum hexacarbonyl	cubane
I	E 12C5 12C52 20C3 15C2	chiral icosahedral or dodecahedral	Rhinovirus	snub dodecahedron	human polio virus
Ih	E 12C5 12C52 20C3 15C2 i 12S10 12S103 20S6 15σ	icosahedral or dodecahedral	Buckminsterfullerene	dodecaborate anion	dodecahedrane

Laue classes

All of the group operations described above and the symbols for crystallographic point groups themselves were first published by Arthur Schoenflies in 1891. Later Max von Laue published the results of experiments using x-ray diffraction to elucidate the internal structures of crystals, producing a limited version of the table of "Laue classes" shown below which are sometimes described as "Laue groups" or "Friedel classes" (after Georges Friedel). Hermann-Mauguin notation is almost invariably used now to describe crystallographic groups in Laue classes but this system provides no advantage for atomic and molecular work.

Point groups in Laue classes

When adapted for molecular work this table first divides point groups into three kinds: asymmetric, symmetric and spherical tops. These are categories based on the angular momentum of molecules, having respectively 3, 2 and 1 distinct values of angular momentum, becoming more symmetrical down the table. A further sub-division into systems is defined by the rotational group G in the leftmost column then into rows of Laue classes that take the form of cyclic and dihedral groups in the first two categories and tetrahedral and octahedral classes in the third.

Rotational groups contain only pure rotational operations, sometimes called proper operations and occur only in the first column of the table. Von Laue showed that x-ray diffraction can not distinguish between groups in a row of the table and shows each one to be the centred point group on the right hand column of the table. Groups in this column contain the inversion operation itself as a member. The middle two columns contain non-rotational groups belonging to the same abstract group as that in the first column -this is why they result in the same diffraction pattern

For example, seven groups in the hexagonal system all contain the C₆ cyclic system, mostly as physical rotational group but in the third column of the table as an abstract group. So, C₆ and C_3h are distinct manifestations of the same group while C_6h is simply C₆ × i. Groups D₆, C_6v and D_3h are also example of the same abstract group and D_6h is the direct product D₆ × i. Tetrahedral and octahedral point groups have a relationship similar to that between cyclic and dihedral groups and the tetrahedral group occurs in all cubic groups.

Representations and their characters

A set of matrices that multiply together in a way that mimics the multiplication table of the elements of a group is called a representation of the group. The simplest method of obtaining a representation of molecular group transformations is to trace the movements of atoms in a molecule when symmetry operations are applied. For example, a water molecule belonging to the C_2v point group might have an oxygen atom labelled 1 and two hydrogen atoms labelled 2 and 3 as shown in the right hand column vector below. If the hydrogen atoms are imagined to rotate by 180 degrees about an axis passing through the oxygen atom we have the familiar C₂ operation of this point group. The oxygen atom in position number 1 stays in position but the atoms in positions 2 and 3 are moved to positions 3 and 2 in the resulting column vector. The matrix connecting the two provides a 3 × 3 representation for this operation.

${\displaystyle \left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\times {\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]}_{}}$

This point group only contains four operations and matrices for the other three operations are obtained similarly, including the identity matrix which just contains 1's on the leading diagonal (top left to bottom right) and 0's elsewhere. Having obtained the representation matrices in this way it is not difficult to show that they multiply out in exactly the same way as the operations themselves.

${\displaystyle \underset{C_2}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]}}\times \underset{{\sigma }_{\text{v}}}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}}=\underset{{\sigma }_{\text{v}}^{\prime }}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]}}}$

Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are all that are needed as all other representations of the group can be described as a direct sum of the irreducible representations. The first step in finding the irreps making up a given representation is to sum up the values of the leading diagonals for each matrix so, taking the identity matrix first then the matrices in the order above, one obtains (3, 1, 3, 1). These values are the traces or characters of the four matrices. Asymmetric point groups such as C_2v only have 1-dimensional irreps so the character of an irrep is exactly the same is the irrep itself and the following table can be interpreted as irreps or characters.

C2v	E	C2	σv(xz)	σv'(yz)
A1	1	1	1	1	z	x2, y2, z2
A2	1	1	−1	−1	Rz	xy
B1	1	−1	1	−1	x, Ry	xz
B2	1	−1	−1	1	y, Rx	yz

Looking again at the characters obtained for the 3D representation above (3, 1, 3, 1), we only need simple arithmetic to break this down into irreps. Clearly, E = 3 means there are three irreps and a C₂ representation sum of 1 means there must be two A and one B irreps so the only combination that adds up to the characters derived is 2A₁+ B₁

Robert Mulliken was the first to publish character tables in English and so the notation used to label irreps in the above table is called Mulliken notation. For asymmetric groups it consists of letters A and B with subscripts 1 and 2 as above and subscripts g and u as in the C_2h example below. (Subscript 3 also appears in D₂) The irreducible representations are those matrix representations in which the matrices are in their most diagonal form possible and for asymmetric groups this means totally diagonal. One further thing to note about the irrep/character table above is the appearance of polar and axial base vector symbols on the right hand side. This tells us that, for example, cartesian base vector × transforms as irrep B₁ under the operations of this group. The same collection of product base vectors is used for all asymmetric groups but symmetric and spherical groups use different sets of product base vectors.

1,5-dibromonapthalene

Point group C_2h has the operations {E, C₂, i, σ_h } and the 1,5-dibromonapthalene (C₁₀H₆Br₂) shown in the figure belongs to this symmetry group. It is possible to construct four 18 × 18 matrices representing the transformations of atoms during its symmetry operations in the style of the water molecule example above and reduce it to 18 1D irreps. Notice however that carbon atom number 1 either stays in place or it is exchanged with carbon atom number 5 and these two atoms can be analysed separately from all the other atoms in the molecule. The transformation matrix for these two atoms alone during the molecular C₂ rotation is

${\displaystyle \left[\begin{array}{c}5\\ 1\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\times {\left[\begin{array}{c}1\\ 5\end{array}\right]}_{}}$

with character 0. When this computation is carried out for each of the operations above the characters obtained are (2,0,0.2) because two operations leave the atoms in place and two move them. The irrep table for this group is below. The first column tells us there are two1D irreps, the second column (C₂) that there is one A and one B while columns 3 and 4 reveal that one irrep has subscript g the other has to have subscript u. This means that the irreps resulting from the two atoms are A_g + B_u. In fact, the 18 atoms in this molecule are paired off in exactly the same way as carbon atoms 1 and 5 so that, from a symmetry perspective, the atom consists of 9 pairs of equivalent atoms related through symmetry. It follows that each pair contributes the same irreps as the pair examined above to give a total 18 dimensional irrep result of 9(A_g + B_u).

C2h	E	C2	i	σh
Ag	1	1	1	1	Rz	xy, x2, y2, z2
Bg	1	-1	1	−1	Rx, Ry	xz,yz
Au	1	1	−1	−1	z
Bu	1	−1	−1	1	x, y

Symmetric point group cyclic irreps

Symmetric point group representations

Symmetric point groups are divided into systems based on the increasing order of the main rotational axis from three to infinity. Systems are in turn divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have one dimensional representations as shown in the table of irreps and the number of irreps is equal to the order of the group. The irreps shown use standard notation for the rotational group of a class but Mulliken sometimes gave different symbols to other members of the same class even though they belong to the same abstract group and therefore have the same irreps.

Symmetric group dihedral irreps

Dihedral point groups contain a cyclic group of the same rotational order: so group D_n always contains group C_n as an index-2 subgroup. It follows that dihedral irreps are superimposed on cyclic irreps because the cyclic group within a dihedral one does not cease to be a cyclic group. A dihedral group also contains a 2-fold rotational axis at right angles to the main cyclic axis and this has two consequences. Firstly, the A and B cyclic irreps are split into pairs of one dimensional irreps identified by subscripts 1 and 2. Secondly, pairs of 1D E_−x and E_+x cyclic irreps combine to form single E_x 2D irreps in the dihedral group because the 2-fold horizontal rotation makes pairs of rotations equivalent. For example, a 60 degree rotation about the main axis becomes equivalent to a (360 − 60) degree rotation because the 2-fold horizontal rotation makes them equivalent. Combinations of this kind are said to form a class. Infinite order dihedral group irreps sometimes use Greek symbol descriptions, Σ, Π, Δ that follow from early linear molecule calculations.

Taking benzene as a simple example, we have a molecule that belongs to the series of point groups C₆, D₆ and D_6h with increasing orders 6, 12 and 24. The six carbon atoms may be represented by a 6 × 6 matrices which in group C₆ have irreps A,B, E₊₁,E_-1,E₊₂ and E_-2 because n objects in an n-fold cyclic group always produce one of each irrep. If these cyclic irreps are promoted to group D₆ we obtain A₁, B_x, E₁ and E₂ when subscripts are added to the 1D A and B irreps and the others merge to form 2D irreps. A₁ is there because the most symmetric irrep has to occur once and only once in the result leaving only the B irrep subscript to be deduced. The character of the 2-fold horizontal rotation operation is 2 because 2 carbon atoms stay in place during the rotation, Char(C₂) = 2 telling us that there are two more 1 subscripts than 2 subscripts so the result is A₁, B₁, E₁ and E₂. Finally, promotion to D_6h requires the addition of g and u subscripts. Since Char(i) = 0 there are an equal number of g and u subscripts, and A_1g has to be present as the most symmetric group so B_1u is mandatory. Furthermore, odd and even 2D irreps take u and g subscripts so the final result for the carbon atoms is (A_1g, B_1u, E_1u, E_2g), but with the hydrogen atoms we get 2(A_1g, B_1u, E_1u , E_2g).

Boric acid and boron trifluoride provide further hexagonal examples in spite of their slightly misleading Schoenflies symbols C_3h and D_3h. Taking boric acid first, we have three sets of equivalent atoms: 1 boron, 3 oxygen and 3 hydrogen. Obviously the oxygen and hydrogen atoms produce the same irreps so only one has to be deduced. Applying the 6-fold cyclic group to (say) the hydrogen atoms produces characters (3,0,0,3,0,0) yielding irreps A + E₊₂ +E_-2. Doubling up and adding an irrep for the central boron atom produces A + 2(A +E₊₂ +E_-2) in standard Laue class notation. Unfortunately, Mulliken used a different notation for C_3h and D_3h irreps to that used for other groups and a conversion table would be needed if that was important.

Boron trifluoride has a central boron atom with 3 fluorine atoms and belongs to cyclic subgroup C_3h and the larger dihedral group D_3h. Following the above reasoning, the irreps in C_3h are A +(A + E₊₂ +E_-2) and when promoted to the dihedral group this becomes A₁ + (A₁ + E₂). Again, conversion to Mulliken notation is required if that is important.

Spherical point group representations

Spherical classes are defined by the tetrahedral, octahedral and icosahedral rotational groups T, O and I. The first two of these, T and O, are related in much the same way as cyclic and dihedral groups are related in symmetric groups. Both tetrahedral and octahedral molecules are often shown with their atoms inscribed in the apices or faces of cubes and might be considered as a single "cubic" system. The first Laue class of this system contains only the tetrahedral rotational group T of order 12 and the direct product of this group with space inversion T_h of order 24. Every point group in the following octahedral class contains the tetrahedral rotational group as a subgroup. Irreps of tetrahedral and octahedral groups are also related similarly to cyclic and dihedral groups and the table below shows how tetrahedral irreps are incorporated in octahedral irreps

Tetrahedral	A	E+, E-	T
Octahedral	A1, A2	E	T1, T2

Tetrahedral symmetry has 3 one dimensional irreps (A, E_{+ ,}E_-) and one 3 dimensional irrep T then the A and T irreps split into two irreps with subscripts 1 and 2 while the two 1D E irreps combine into a single 2D irrep. Notice that the T irrep is always 3 dimensional but the E irrep only becomes 2 dimensional in the higher order group.

Methane (CH₄) is often used as an example and, although often described as a tetrahedral molecule because of the very visible rotational symmetry, it really belongs to the octahedral symmetry class. Considering methane first as a tetrahedral molecule the 12 operations of group T are {E, 3 × c, 4 × b, 4 × b³} where c is a 180 degree rotation along x, y and z axes and b is a 120 degree rotation about the apices of a cube. It is not difficult to convert 5 × 5 symmetry operation transformation matrices to reducible matrices and thence to molecular irreps but this not necessary.

Methane has two sets of equivalent atoms: a single carbon atom and 4 hydrogen atoms. The atoms of each set are transformed into each other during operations. A single atom can only ever be transformed into itself and therefore always contributes the most symmetrical irrep to the end total irrep count. Additionally, there is a rule of group theory that the most symmetrical irrep must occur once and only once in the irreps of any equivalent atom set so the five dimensions of irreps being sought contain 2A and three others. Although E₁ and E₂ are 1 dimensional they have to occur together in the irreps of any equivalent set of atoms. it follows that he only way of filling the remaining three dimensions is to adopt 3D irrep T so the irreps are 2A + T. (E irreps have to be taken in pairs in physical molecular applications).

               methane                        sulfur hexafluoride

Extending this treatment to the octahedral group T_d requires six 4-fold roto-inversion operations (f) about the main axes and six 2-fold roto-inversions (a), appearing as mirror reflections through opposite edges of the imaginary cube in which methane is placed. So half the operations of this group are rotational and half non-rotational. Rotational group T exists within the non-rotational group T_d = {E, 3 × c, 8 × b/b³, 6 × f, 6 × a} so the irreps in group T in the expansion to T_d. Again we have 2 sets of equivalent atoms and each set must contribute one and only one of the most symmetrical irrep, in this case A₁. Reasoning as above, we know that the irreps in T_d must be 2A₁ + T_x so the last step is to find the 3D subscript. A brief look at the 4 × 4 transformation matrix for the 4-fold rotation operation f shows character Ch(f) = 0 and the subscript × has to be 2 to balance the 1 on the A irrep. so the final result is 2A₁ + T₂

Sulfur hexafluoride (SF₆) can also be treated first as a tetrahedral molecule T, then as octahedral O and finally as centred molecule O_i. There are two sets of equivalent atoms consisting of a single sulfur atom and six fluorine atoms. Transformations of the fluorine atoms generate a six dimensional representation that can only reduce into the direct sum of tetrahedral irreps A, E₊₁, E_-1 and T because the direct sum must include the most symmetrical irrep once and only once, leaving five dimensions that can only be satisfied in the way shown - a direct sum of 5 can only be made up from a 2 and a 3 - no other combination is possible. These irreps are "promoted" to 2A₁ + E₁ + T_x in group O. To get the × subscript observe that the 4-fold rotation in SF₆ has character Ch(f) = 2 because two atoms stay in position and a glance at this column of the table suggests A₁ + E₁ + T₁. Finally the inversion operation (i) applied go the fluorine atoms has character Ch(i) = 0 indicating equal numbers of g and u subscripts (because none of the atoms remains in position). Since the most symmetrical irrep must occur once the only possible result is A_1g + E_1g + T_1u. The single sulfur atom always has the most symmetric irrep to the final reduction of the seven dimensional matrices to a direct sum is 2A_1g + E_1g + T_1u.

A summary of possible point group irreducible representations

Cyclic groups have Schoenflies symbols C_n, S_n, C_nh, C_s and C_i positioned together in rows of the Laue class table above. These point groups are represented by the one dimensional symbols A, B, E_+x and E_-x. If the group has a centre of symmetry the number if irreps doubles and subscripts g and u have to be added to each of the simple cyclic symbols. For example, the eight possible irreps of C_4h are A_g, B_g, E_+1g, E_-1g, A_u, B_u, E_+1u and E_-1u. Cotton (page 96) shows how cyclic group irreps are derived for groups of any order.

Dihedral groups have Schoenflies symbols D_n, C_nv, D_nd and D_nh also positioned together in rows of the Laue class table. They are represented by one and two dimensional irreps derived from the cyclic irreps above. Cyclic A and B irreps split into separate one dimensional irreps A₁, A₂, B₁ and B₂ while pairs of E_+x and E_-x irreps merge to form single two dimensional degenerate irreps. For example, the possible irreps of D₄ (also C_4v and D_2d) are A₁, A₂, B₁,B₂ and E. (E₁ is normally shown simply as E). A group with a centre of symmetry also has irreps with g and u subscripts. Once the irreps of a cyclic group C_n are known it is a simple task to deduce the irreps of the corresponding dihedral group D_n.

In addition to the one and two dimensional irreps so far described, tetrahedral groups can a three dimensional degenerate irrep T that expands in octahedral groups to yield irreps T₁ and T₂.

The following reference of character tables uses symbols Z_x for abstract cyclic groups C_x with A₄ and S₄ (alternating and symmetric permutations of 4 objects) for T and O. Many authors just use C, T and O in two senses, making it clear which is intended.

Main article: List of character tables for chemically important 3D point groups

Historical background

See also: Chemical crystallography before X-rays § Molecular chirality

The article above provides an insight into the development of symmetry theory in the context of crystallographic investigations during the 19th century. Point groups were derived from observations of the macroscopic forms of crystals, leading up to the Schoenflies system used to describe them. Max von Laue's invention and use of x-ray diffraction to elucidate the internal structure provided an insight not possible from an examination of external crystal shapes. Laue showed that the 32 crystallographic point groups can be collected into 11 classes each containing 2,3 or 4 point groups that all produce the same diffraction pattern. Crystallographic Laue classes are now very familiar and, although usually shown in international notation, would probably originally have been published in Schoenflies notation. (International notation was developed 10 years after Laue's work). Each class contains one rotational group, one non-rotational group that is the direct product of the rotational group with space inversion and 0,1 or 2 more groups obtained from combinations of rotational group operations with space inversion. Of course, the crystallographic restriction applies only to crystals and in the wider field of molecular symmetry an infinite number of Laue classes become possible.

While crystallographic research continued apace in the first part of the 20th century there were also major discoveries at the atomic level. Rutherford showed that a single atom consists of a massive nucleus surrounded mainly by empty space containing electrons, leading directly to the planetary Bohr model of the atom. Planck discovered that black body radiation could be explained if he allocated particles to discrete energy levels then Einstein showed that the energy levels were characteristic of particles themselves. De Broglie extended this reasoning to all particles, providing a relationship between the allowed energy levels and the wave nature of the particle. All of this work on the combination of Newtonian particle particle behaviour combined with the wave nature of the particles led up to Schrodinger's three dimensional wave equation, perhaps the most fundamental mathematical expression in modern chemistry. Solutions to this equation are defined by 3 quantum numbers that label the allowed energy levels when it is applied to particles, The equation might be applied to any quantized state.

A particle in energy level E₁ might be stimulated to rise to level E₂ by the absorption of a photon with energy ε provided that

E₂ - E₁ = ε

or a particle might fall from the higher energy level to the lower one and emit a photon with the same energy. Like all particles the photon has an energy related to its frequency ε = hν where h is Planck's constant (6.626 × 10⁻³⁴ JHz⁻¹) and ν is the frequency. Higher frequencies have greater energies. This is the basis of "spectroscopy", a procedure in which the energy levels in atoms and molecules probed by photons of varying frequency until the frequency matches an energy level difference. These techniques may be applied in the following three areas

• electronic transitions in atoms and molecules that have frequencies in the range 430 to 770 THz
• vibrational transitions in molecules with typical frequencies 10 to 100 THz
• rotational transitions of molecules with common frequencies in the range 8 to 10 GHz

Clearly, these frequencies are in the order electronic > vibrational > rotational so the energy differences also change in this order. Suppose an atom or molecule emits or absorbs at 500 THz, a quick calculation shows the energy involved to be

ε = hν = (6.626 × 10⁻³⁴) × (500 × 10¹²) = 3.313 × 10⁻¹⁹ J

and it becomes clear that at the atomic level levels involved here are incredibly small in relation to day to day experience.

Soon after the Schrodinger equation became familiar to early researchers Hans Bethe showed how atomic orbitals could be modified by symmetric "crystal fields" resulting from surrounding charges and this work was extended to his study of ligand field theory in 1929. Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy.

E. Bright Wilson used character tables in 1934 to predict the symmetry of vibrational normal modes.

Electronic and vibrational spectra provide enormous detail about molecular structures and, although rotational spectra cannot be linked to individual point groups, it does often supply useful information.

Symmetry of molecular orbitals

When Schrodinger's 3D wave equation is applied to a one-electron atom it provides a number of solutions called wave functions that are then used to label the allowed energy levels in that atom. Exact solutions of this kind are usually described by three quantum numbers, n, l and m from which the probable radial and angular distribution of the electron around the atom can be computed. This type of deduction leads to the familiar s, p, d, f, ... description of atomic orbitals based on the l and m quantum numbers. Each solution is a base vector from which more complex structures may be constructed. Descriptions of many-electron atoms use the one-electron model to build models that are sometimes pictured as multiple electrons in the simple structure. Molecular orbitals then take linear combinations of atomic orbitals (LCAOs) to explain the distribution of electrons over multiple atoms within a molecule. Atomic orbital symmetry follows from the angular part of the wave function which increases in complexity in the series s,p,d,f,... so that s orbitals only have radial symmetry while p orbital base vectors have a symmetry identical to that of the Cartesian polar base vectors.

Consider the example of water (H₂O), which has the C_2v symmetry described above. The 2p_x orbital of oxygen has, like the × base vector, B₁ symmetry. It is oriented perpendicular to the plane of the molecule and switches sign with a C₂ and a σ_v'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B₁ irreducible representation. Likewise, the 2p_z orbital is seen to have the symmetry of the A₁ irreducible representation (i.e.: none of the symmetry operations change it), 2p_y B₂, and the 3d_xy orbital A₂. These assignments are noted in the rightmost columns of the table.

Each molecular orbital also has the symmetry of one irreducible representation. For example, ethylene (C₂H₄) has symmetry group D_2h, and its highest occupied molecular orbital (HOMO) is the bonding pi orbital which forms a basis for its irreducible representation B_1u.

Symmetry of vibrational modes

Electronic bonds between atoms in molecules can be imagined to be equivalent to springs that extend, contract and bend, absorbing energies appropriate for the strength of the bond. Each of the N atoms in a molecule can move to a limited extent in three dimensions so that a total of 3N distinct motions become possible but 3 of these correspond to the overall translational motion of the molecule and 3 to overall rotation. It follows that there are 3N - 6 genuine motions relative to the whole semi-rigid molecule. (3N -5 in linear molecules). Vibrational motions of this kind may be resolved into normal modes of molecular vibration in which each mode has a symmetry which forms a basis for one irreducible representation of the molecular symmetry group. This relationship is the basis of vibrational spectroscopy. Irreps for the normal modes of a molecule are derived through a fairly mechanical procedure

1. find the point group of the molecule
2. deduce the irreps for that molecule
3. look up the irreps corresponding to x, y and z base vectors for an atom in the point group
4. take the direct product of the molecular irreps and the three base vector irreps
5. subtract the six irreps that represent overall rotations and translations

Vibrational modes for a water molecule

A water molecule (H₂O) has three atoms and so has 3 × 3 - 6 = 3 normal modes of vibration. The molecular symmetry of water is C_2v and the molecular irreps 2A₁ + B₂. A glance at the irrep/character table shows that the x, y and z base vectors for C_2v are A₁, B₁ and B₂. The direct product is then

(2A₁ + B₂) × (A₁ + B₁+ B₂) = 2(A₁ + B₁+ B₂) + (B₂ + A₂ + A₁) = 3A₁ + A₂ + 2B₁+ 3B₂

There are three atoms in the molecule so the resulting representation is 3N = 3 × 3 = 9 dimensional. Subtracting the three polar base vectors A₁, B₁ and B₂ and the axial base vectors B₂, B₁ and A₁ a final result is obtained

(3A₁ + A₂ + 2B₁+ 3B₂) - ((A₁, B_1, B₂) + ( B₂, B₁, A₁)) = 2A₁ + B₂

The overall symmetry of the three vibrational modes is therefore 2A₁ + B₂.^[24][25]

The three modes represents a symmetric stretch in which the two O-H bond lengths vary in phase with each other, an asymmetric stretch in which they vary out of phase, and a bending mode in which the bond angle varies. Symmetric stretching and the bending modes have symmetry A₁, while the asymmetric mode has symmetry B₂

Diborane

Diborane (B₂H₆ ) has D_2h molecular symmetry. The terminal B-H stretching vibrations which are active in IR are B_2u and B_3u.

Vibrational modes for an ammonia molecule

An ammonia molecule (NH₃) has 4 atoms and therefore 3N - 6 = 6 normal modes of vibration. It has a trigonal pyramidal shape with 3 hydrogen atoms equidistant from a nitrogen atom and belongs to point group C_3v, with symmetry operations E, C₃ and σ_v. Both of the examples above are asymmetric molecules in which only 1-dimensional irreps are possible and in which each irrep corresponds to a single vibrational mode related to a single frequency. Ammonia is a symmetric molecule and therefore its vibrational modes may include 2-dimensional irreps representing two modes of the same frequency (degenerate modes). Symmetry group C_3v for has the three symmetry species A₁, A₂ and E₁ although the last of these is usually written simply as E.

A brief glance at the molecule shows that the nitrogen atom contributes irrep A₁ while the three hydrogen atoms contribute A₁ + E so the molecular irreps are 2A₁ + E. The translational modes for this point group are A₁ + E and so the direct product of these three vectors with the molecular irreps produces a 12 dimensional representation

(2A₁ + E) × ( A₁ + E) = 3A₁ + A₂ + 4E

Now the 3 translational modes A₁ + E and 3 rotational modes A₂ + E have to be subtracted from this representation

(3A₁ + A₂ + 4E) - (A₂ + E) - (A₁ + E) = 2A₁ + 2E

Ammonia molecule

All three hydrogen atoms travel symmetrically along the N-H bonds, either in the direction of the nitrogen atom or away from it. This mode is known as symmetric stretch (v₁) and reflects the symmetry in the N-H bond stretching. Of the three vibrational modes, this one has the highest frequency.

In the Bending (ν₂) vibration, the nitrogen atom stays on the axis of symmetry, while the three hydrogen atoms move in different directions from one another, leading to changes in the bond angles. The hydrogen atoms move like an umbrella, so this mode is often referred to as the "umbrella mode".

There is also an Asymmetric Stretch mode (ν₃) in which one hydrogen atom approaches the nitrogen atom while the other two hydrogens move away.

Vibrational modes for spherical molecules

Molecules such as methane and carbon tetrachloride with a general formula AB₄ have a very obvious tetrahedral appearance and are usually described as such even though they belong to the octahedral Laue class. They have five atoms: a central atom A surrounded by four equivalent B atoms so we might expect (5 × 3) - 6 = 9 vibrational modes. Molecular irreps for this shape were earlier found to be 2A₁ + T₂ and a direct product of this five dimensional expression with the irrep representing x, y and z base vector transformations (T₂ ) produces a 15 dimensional result

(2A₁ + T₂) × T₂ = (2T₂ + (A₁ + E + T₁ + T₂)) = A₁ + E + T₁ + 3T₂

A subtraction of the polar and axial base vectors (T₂ and T₁) produces a final 9 dimensional result

A₁ + E + T₁ + 3T₂ - (T₁ + T₂) = A₁ + E + 2T₂

All three of these modes are Raman active but only T₂ is IR active.

While there are simple rules to deduce the results of direct products for asymmetric and symmetric groups the results for spherical molecules are best found from tables. For example the product above used the identity T₂ × T₂ = A₁ + E + T₁ + T₂. This is equally applicable to other octahedral groups because O and Td are distinct examples of the same abstract group.

W(CO)₆ has octahedral geometry. The irreducible representations contributing to the six dimensional representation for the C-O stretching vibration are A_1g + E_g + T_1u . Of these, only T_1u is IR active. ???

Symmetry and molecular rotation

Rotational molecule moments of inertia

Molecular rotation is quantized but the relationship between molecular symmetry and allowed energy levels is not as detailed as that for electronic and vibrational motions. Rotational transitions depend only on the kinetic energy of the rotating molecule while electronic and vibrational levels involve both kinetic and potential energies. In spite of this difference the general procedure for relating molecular structure to observed spectra is similar to that of the earlier systems. First of all an expression is derived for the energy of the rotating molecule, taking into account the Laue partition to which it belongs because rotational motion is not related to individual point groups or even to their class. Asymmetric molecules have three different moments of inertia from which three different angular momenta can be derived: L_x, L_y and L_z. The fact that there are three distinct values of angular momentum means that solutions to the Schrodinger for this kind of molecule are exceedingly difficult. This does not stop asymmetric molecules exhibiting clear rotational spectra: water vapour provides clear transition lines.

Symmetric molecules have two equal moments of inertia and so two equal angular momenta, making their energy expressions more easily soluble. Oblate symmetric molecules have a discus shape (e.g. benzene) and prolate examples are cigar shaped (e.g. methyl chloride). Prolate symmetric molecules become linear when the main axial order reaches infinity at which point there is only one moment of inertia and one angular momentum. This makes the Schrodinger equation easily soluble and it is possible to relate (say) diatomic molecule structure to their structure very accurately. Spherical molecules also only have one moment of inertia so their energy levels are easily calculated but not always observed.

Point group symmetry describes the symmetry of a molecule when fixed at its equilibrium configuration in a particular electronic state. It does not allow for tunneling between minima nor for the change in shape that can come about from the centrifugal distortion effects of molecular rotation.

The molecular symmetry group

One can determine the symmetry operations of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one uses a point group to classify molecular states, the operations in it are not to be interpreted in the same way. Instead the operations are interpreted as rotating and/or reflecting the vibronic (vibration-electronic) coordinates and these operations commute with the vibronic Hamiltonian. They are "symmetry operations" for that vibronic Hamiltonian. The point group is used to classify by symmetry the vibronic eigenstates of a rigid molecule. The symmetry classification of the rotational levels, the eigenstates of the full (rotation-vibration-electronic) Hamiltonian, can be achieved through the use of the appropriate permutation-inversion group (called the molecular symmetry group), as introduced by Longuet-Higgins.

Molecular nonrigidity

See also: Fluxional molecule

As discussed above in § The molecular symmetry group, point groups are useful for classifying the vibrational and electronic states of rigid molecules (sometimes called semi-rigid molecules) which undergo only small oscillations about a single equilibrium geometry. Longuet-Higgins introduced the molecular symmetry group (a more general type of symmetry group) suitable not only for classifying the vibrational and electronic states of rigid molecules but also for classifying their rotational and nuclear spin states. Further, such groups can be used to classify the states of non-rigid (or fluxional) molecules that tunnel between equivalent geometries and to allow for the distorting effects of molecular rotation. The symmetry operations in the molecular symmetry group are so-called 'feasible' permutations of identical nuclei, or inversion with respect to the center of mass (the parity operation), or a combination of the two, so that the group is sometimes called a "permutation-inversion group".

Examples of molecular nonrigidity abound. For example, ethane (C₂H₆) has three equivalent staggered conformations. Tunneling between the conformations occurs at ordinary temperatures by internal rotation of one methyl group relative to the other. This is not a rotation of the entire molecule about the C₃ axis, although each conformation has D_3d symmetry, as in the table above. The molecule 2-butyne (dimethylacetylene) has the same molecular symmetry group (G₃₆) as ethane but a very much lower torsional barrier. Similarly, ammonia (NH₃) has two equivalent pyramidal (C_3v) conformations which are interconverted by the process known as nitrogen inversion.

Additionally, the methane molecule (CH₄) and trihydrogen cation (H₃⁺) have highly symmetric equilibrium structures with T_d and D_3h point group symmetries respectively; they lack permanent electric dipole moments but they do have very weak pure rotation spectra because of rotational centrifugal distortion.

Sometimes it is necessary to consider together electronic states having different point group symmetries at equilibrium. For example, in its ground (N) electronic state the ethylene molecule C₂H₄ has D_2h point group symmetry whereas in the excited (V) state it has D_2d symmetry. To treat these two states together it is necessary to allow torsion and to use the double group of the molecular symmetry group G₁₆.