Von Neumann architecture

From Wikipedia, the free encyclopedia

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

A von Neumann architecture scheme

The von Neumann architecture — also known as the von Neumann model or Princeton architecture — is a computer architecture based on a 1945 description by John von Neumann, and by others, in the First Draft of a Report on the EDVAC. The document describes a design architecture for an electronic digital computer with these components:

• A processing unit with both an arithmetic logic unit and processor registers
• A control unit that includes an instruction register and a program counter
• Memory that stores data and instructions
• External mass storage
• Input and output mechanisms

The term "von Neumann architecture" has evolved to refer to any stored-program computer in which an instruction fetch and a data operation cannot occur at the same time (since they share a common bus). This is referred to as the von Neumann bottleneck, which often limits the performance of the corresponding system.

The design of a von Neumann architecture machine is simpler than in a Harvard architecture machine—which is also a stored-program system, yet has one dedicated set of address and data buses for reading and writing to memory, and another set of address and data buses to fetch instructions.

A stored-program digital computer keeps both program instructions and data in read–write, random-access memory (RAM). Stored-program computers were an advancement over the program-controlled computers of the 1940s, such as the Colossus and the ENIAC. Those were programmed by setting switches and inserting patch cables to route data and control signals between various functional units. The vast majority of modern computers use the same memory for both data and program instructions, but have caches between the CPU and memory, and, for the caches closest to the CPU, have separate caches for instructions and data, so that most instruction and data fetches use separate buses (split cache architecture).

History

The earliest computing machines had fixed programs. Some very simple computers still use this design, either for simplicity or training purposes. For example, a desk calculator (in principle) is a fixed program computer. It can do basic mathematics, but it cannot run a word processor or games. Changing the program of a fixed-program machine requires rewiring, restructuring, or redesigning the machine. The earliest computers were not so much "programmed" as "designed" for a particular task. "Reprogramming" – when possible at all – was a laborious process that started with flowcharts and paper notes, followed by detailed engineering designs, and then the often-arduous process of physically rewiring and rebuilding the machine. It could take three weeks to set up and debug a program on ENIAC.

With the proposal of the stored-program computer, this changed. A stored-program computer includes, by design, an instruction set, and can store in memory a set of instructions (a program) that details the computation.

A stored-program design also allows for self-modifying code. One early motivation for such a facility was the need for a program to increment or otherwise modify the address portion of instructions, which operators had to do manually in early designs. This became less important when index registers and indirect addressing became usual features of machine architecture. Another use was to embed frequently used data in the instruction stream using immediate addressing. Self-modifying code has largely fallen out of favor, since it is usually hard to understand and debug, as well as being inefficient under modern processor pipelining and caching schemes.

Capabilities

On a large scale, the ability to treat instructions as data is what makes assemblers, compilers, linkers, loaders, and other automated programming tools possible. It makes "programs that write programs" possible. This has made a sophisticated self-hosting computing ecosystem flourish around von Neumann architecture machines.

Some high level languages leverage the von Neumann architecture by providing an abstract, machine-independent way to manipulate executable code at runtime (e.g., LISP), or by using runtime information to tune just-in-time compilation (e.g. languages hosted on the Java virtual machine, or languages embedded in web browsers).

On a smaller scale, some repetitive operations such as BITBLT or pixel and vertex shaders can be accelerated on general purpose processors with just-in-time compilation techniques. This is one use of self-modifying code that has remained popular.

Development of the stored-program concept

The mathematician Alan Turing, who had been alerted to a problem of mathematical logic by the lectures of Max Newman at the University of Cambridge, wrote a paper in 1936 entitled On Computable Numbers, with an Application to the Entscheidungsproblem, which was published in the Proceedings of the London Mathematical Society. In it he described a hypothetical machine he called a universal computing machine, now known as the "Universal Turing machine". The hypothetical machine had an infinite store (memory in today's terminology) that contained both instructions and data. John von Neumann became acquainted with Turing while he was a visiting professor at Cambridge in 1935, and also during Turing's PhD year at the Institute for Advanced Study in Princeton, New Jersey during 1936–1937. Whether he knew of Turing's paper of 1936 at that time is not clear.

In 1936, Konrad Zuse also anticipated, in two patent applications, that machine instructions could be stored in the same storage used for data.

Independently, J. Presper Eckert and John Mauchly, who were developing the ENIAC at the Moore School of Electrical Engineering of the University of Pennsylvania, wrote about the stored-program concept in December 1943. In planning a new machine, EDVAC, Eckert wrote in January 1944 that they would store data and programs in a new addressable memory device, a mercury metal delay-line memory. This was the first time the construction of a practical stored-program machine was proposed. At that time, he and Mauchly were not aware of Turing's work.

Von Neumann was involved in the Manhattan Project at the Los Alamos National Laboratory. It required huge amounts of calculation, and thus drew him to the ENIAC project, during the summer of 1944. There he joined the ongoing discussions on the design of this stored-program computer, the EDVAC. As part of that group, he wrote up a description titled First Draft of a Report on the EDVAC based on the work of Eckert and Mauchly. It was unfinished when his colleague Herman Goldstine circulated it, and bore only von Neumann's name (to the consternation of Eckert and Mauchly). The paper was read by dozens of von Neumann's colleagues in America and Europe, and influenced the next round of computer designs.

Jack Copeland considers that it is "historically inappropriate to refer to electronic stored-program digital computers as 'von Neumann machines'". His Los Alamos colleague Stan Frankel said of von Neumann's regard for Turing's ideas

I know that in or about 1943 or '44 von Neumann was well aware of the fundamental importance of Turing's paper of 1936…. Von Neumann introduced me to that paper and at his urging I studied it with care. Many people have acclaimed von Neumann as the "father of the computer" (in a modern sense of the term) but I am sure that he would never have made that mistake himself. He might well be called the midwife, perhaps, but he firmly emphasized to me, and to others I am sure, that the fundamental conception is owing to Turing— in so far as not anticipated by Babbage…. Both Turing and von Neumann, of course, also made substantial contributions to the "reduction to practice" of these concepts but I would not regard these as comparable in importance with the introduction and explication of the concept of a computer able to store in its memory its program of activities and of modifying that program in the course of these activities.

At the time that the "First Draft" report was circulated, Turing was producing a report entitled Proposed Electronic Calculator. It described in engineering and programming detail, his idea of a machine he called the Automatic Computing Engine (ACE). He presented this to the executive committee of the British National Physical Laboratory on February 19, 1946. Although Turing knew from his wartime experience at Bletchley Park that what he proposed was feasible, the secrecy surrounding Colossus, that was subsequently maintained for several decades, prevented him from saying so. Various successful implementations of the ACE design were produced.

Both von Neumann's and Turing's papers described stored-program computers, but von Neumann's earlier paper achieved greater circulation and the computer architecture it outlined became known as the "von Neumann architecture". In the 1953 publication Faster than Thought: A Symposium on Digital Computing Machines (edited by B. V. Bowden), a section in the chapter on Computers in America reads as follows:

The Machine of the Institute For Advanced Studies, Princeton

In 1945, Professor J. von Neumann, who was then working at the Moore School of Engineering in Philadelphia, where the E.N.I.A.C. had been built, issued on behalf of a group of his co-workers, a report on the logical design of digital computers. The report contained a detailed proposal for the design of the machine that has since become known as the E.D.V.A.C. (electronic discrete variable automatic computer). This machine has only recently been completed in America, but the von Neumann report inspired the construction of the E.D.S.A.C. (electronic delay-storage automatic calculator) in Cambridge (see page 130).

In 1947, Burks, Goldstine and von Neumann published another report that outlined the design of another type of machine (a parallel machine this time) that would be exceedingly fast, capable perhaps of 20,000 operations per second. They pointed out that the outstanding problem in constructing such a machine was the development of suitable memory with instantaneously accessible contents. At first they suggested using a special vacuum tube—called the "Selectron"—which the Princeton Laboratories of RCA had invented. These tubes were expensive and difficult to make, so von Neumann subsequently decided to build a machine based on the Williams memory. This machine—completed in June, 1952 in Princeton—has become popularly known as the Maniac. The design of this machine inspired at least half a dozen machines now being built in America, all known affectionately as "Johniacs".

In the same book, the first two paragraphs of a chapter on ACE read as follows:

Automatic Computation at the National Physical Laboratory

One of the most modern digital computers which embodies developments and improvements in the technique of automatic electronic computing was recently demonstrated at the National Physical Laboratory, Teddington, where it has been designed and built by a small team of mathematicians and electronics research engineers on the staff of the Laboratory, assisted by a number of production engineers from the English Electric Company, Limited. The equipment so far erected at the Laboratory is only the pilot model of a much larger installation which will be known as the Automatic Computing Engine, but although comparatively small in bulk and containing only about 800 thermionic valves, as can be judged from Plates XII, XIII and XIV, it is an extremely rapid and versatile calculating machine.

The basic concepts and abstract principles of computation by a machine were formulated by Dr. A. M. Turing, F.R.S., in a paper¹. read before the London Mathematical Society in 1936, but work on such machines in Britain was delayed by the war. In 1945, however, an examination of the problems was made at the National Physical Laboratory by Mr. J. R. Womersley, then superintendent of the Mathematics Division of the Laboratory. He was joined by Dr. Turing and a small staff of specialists, and, by 1947, the preliminary planning was sufficiently advanced to warrant the establishment of the special group already mentioned. In April, 1948, the latter became the Electronics Section of the Laboratory, under the charge of Mr. F. M. Colebrook.

Early von Neumann-architecture computers

The First Draft described a design that was used by many universities and corporations to construct their computers. Among these various computers, only ILLIAC and ORDVAC had compatible instruction sets.

• ARC2 (Birkbeck, University of London) officially came online on May 12, 1948.
• Manchester Baby (Victoria University of Manchester, England) made its first successful run of a stored program on June 21, 1948.
• EDSAC (University of Cambridge, England) was the first practical stored-program electronic computer (May 1949)
• Manchester Mark 1 (University of Manchester, England) Developed from the Baby (June 1949)
• CSIRAC (Council for Scientific and Industrial Research) Australia (November 1949)
• MESM in Kyiv, Ukraine (November 1950)
• EDVAC (Ballistic Research Laboratory, Computing Laboratory at Aberdeen Proving Ground 1951)
• ORDVAC (U-Illinois) at Aberdeen Proving Ground, Maryland (completed November 1951)
• IAS machine at Princeton University (January 1952)
• MANIAC I at Los Alamos Scientific Laboratory (March 1952)
• ILLIAC at the University of Illinois, (September 1952)
• BESM-1 in Moscow (1952)
• AVIDAC at Argonne National Laboratory (1953)
• ORACLE at Oak Ridge National Laboratory (June 1953)
• BESK in Stockholm (1953)
• JOHNNIAC at RAND Corporation (January 1954)
• DASK in Denmark (1955)
• WEIZAC at the Weizmann Institute of Science in Rehovot, Israel (1955)
• PERM in Munich (1956)
• SILLIAC in Sydney (1956)

Early stored-program computers

The date information in the following chronology is difficult to put into proper order. Some dates are for first running a test program, some dates are the first time the computer was demonstrated or completed, and some dates are for the first delivery or installation.

• The IBM SSEC had the ability to treat instructions as data, and was publicly demonstrated on January 27, 1948. This ability was claimed in a US patent. However it was partially electromechanical, not fully electronic. In practice, instructions were read from paper tape due to its limited memory.
• The ARC2 developed by Andrew Booth and Kathleen Booth at Birkbeck, University of London officially came online on May 12, 1948. It featured the first rotating drum storage device.
• The Manchester Baby was the first fully electronic computer to run a stored program. It ran a factoring program for 52 minutes on June 21, 1948, after running a simple division program and a program to show that two numbers were relatively prime.
• The ENIAC was modified to run as a primitive read-only stored-program computer (using the Function Tables for program ROM) and was demonstrated as such on September 16, 1948, running a program by Adele Goldstine for von Neumann.
• The BINAC ran some test programs in February, March, and April 1949, although was not completed until September 1949.
• The Manchester Mark 1 developed from the Baby project. An intermediate version of the Mark 1 was available to run programs in April 1949, but was not completed until October 1949.
• The EDSAC ran its first program on May 6, 1949.
• The EDVAC was delivered in August 1949, but it had problems that kept it from being put into regular operation until 1951.
• The CSIR Mk I ran its first program in November 1949.
• The SEAC was demonstrated in April 1950.
• The Pilot ACE ran its first program on May 10, 1950, and was demonstrated in December 1950.
• The SWAC was completed in July 1950.
• The Whirlwind was completed in December 1950 and was in actual use in April 1951.
• The first ERA Atlas (later the commercial ERA 1101/UNIVAC 1101) was installed in December 1950.

Evolution

Single system bus evolution of the architecture

Through the decades of the 1960s and 1970s computers generally became both smaller and faster, which led to evolutions in their architecture. For example, memory-mapped I/O lets input and output devices be treated the same as memory. A single system bus could be used to provide a modular system with lower cost. This is sometimes called a "streamlining" of the architecture. In subsequent decades, simple microcontrollers would sometimes omit features of the model to lower cost and size. Larger computers added features for higher performance.

Design limitations

Von Neumann bottleneck

The shared bus between the program memory and data memory leads to the von Neumann bottleneck, the limited throughput (data transfer rate) between the central processing unit (CPU) and memory compared to the amount of memory. Because the single bus can only access one of the two classes of memory at a time, throughput is lower than the rate at which the CPU can work. This seriously limits the effective processing speed when the CPU is required to perform minimal processing on large amounts of data. The CPU is continually forced to wait for needed data to move to or from memory. Since CPU speed and memory size have increased much faster than the throughput between them, the bottleneck has become more of a problem, a problem whose severity increases with every new generation of CPU.

The von Neumann bottleneck was described by John Backus in his 1977 ACM Turing Award lecture. According to Backus:

Surely there must be a less primitive way of making big changes in the store than by pushing vast numbers of words back and forth through the von Neumann bottleneck. Not only is this tube a literal bottleneck for the data traffic of a problem, but, more importantly, it is an intellectual bottleneck that has kept us tied to word-at-a-time thinking instead of encouraging us to think in terms of the larger conceptual units of the task at hand. Thus programming is basically planning and detailing the enormous traffic of words through the von Neumann bottleneck, and much of that traffic concerns not significant data itself, but where to find it.

Mitigations

There are several known methods for mitigating the Von Neumann performance bottleneck. For example, the following all can improve performance:

• Providing a cache between the CPU and the main memory
• providing separate caches or separate access paths for data and instructions (the so-called Modified Harvard architecture)
• using branch predictor algorithms and logic
• providing a limited CPU stack or other on-chip scratchpad memory to reduce memory access
• Implementing the CPU and the memory hierarchy as a system on chip, providing greater locality of reference and thus reducing latency and increasing throughput between processor registers and main memory

The problem can also be sidestepped somewhat by using parallel computing, using for example the non-uniform memory access (NUMA) architecture—this approach is commonly employed by supercomputers. It is less clear whether the intellectual bottleneck that Backus criticized has changed much since 1977. Backus's proposed solution has not had a major influence. Modern functional programming and object-oriented programming are much less geared towards "pushing vast numbers of words back and forth" than earlier languages like FORTRAN were, but internally, that is still what computers spend much of their time doing, even highly parallel supercomputers.

As of 1996, a database benchmark study found that three out of four CPU cycles were spent waiting for memory. Researchers expect that increasing the number of simultaneous instruction streams with multithreading or single-chip multiprocessing will make this bottleneck even worse. In the context of multi-core processors, additional overhead is required to maintain cache coherence between processors and threads.

Self-modifying code

Main article: Self-modifying code

Aside from the von Neumann bottleneck, program modifications can be quite harmful, either by accident or design. In some simple stored-program computer designs, a malfunctioning program can damage itself, other programs, or the operating system, possibly leading to a computer crash. Memory protection and other forms of access control can usually protect against both accidental and malicious program changes.

Viral phenomenon

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Viral_phenomenon

Viral phenomena are objects or patterns that are able to replicate themselves or convert other objects into copies of themselves when these objects are exposed to them. Analogous to the way in which viruses propagate, the term viral pertains to a video, image, or written content spreading to numerous online users within a short time period. This concept has become a common way to describe how thoughts, information, and trends move into and through a human population.

The popularity of viral media has been fueled by the rapid rise of social network sites, wherein audiences—who are metaphorically described as experiencing "infection" and "contamination"—play as passive carriers rather than an active role to 'spread' content, making such content "go viral". The term viral media differs from spreadable media as the latter refers to the potential of content to become viral. Memes are one known example of informational viral patterns.

History

Terminology

Meme

The word meme was coined by Richard Dawkins in his 1976 book The Selfish Gene as an attempt to explain memetics; or, how ideas replicate, mutate, and evolve. When asked to assess this comparison, Lauren Ancel Meyers, a biology professor at the University of Texas, stated that "memes spread through online social networks similarly to the way diseases do through offline populations." This dispersion of cultural movements is shown through the spread of memes online, especially when seemingly innocuous or trivial trends spread and die in rapid fashion.

Viral

The term viral pertains to a video, image, or written content spreading to numerous online users within a short time period. If something goes viral, many people discuss it. Accordingly, Tony D. Sampson defines viral phenomena as spreadable accumulations of events, objects, and affects that are overall content built up by popular discourses surrounding network culture. There is also a relationship to the biological notion of disease spread and epidemiology. In this context, "going viral" is similar to an epidemic spread, which occurs if more than one person is infected by a disease for every person infected. Thus, if a piece of content is shared with more than one person every time it is seen, then this will result in viral growth.

In Understanding Media (1964), philosopher Marshall McLuhan describes photography in particular, and technology in general, as having a potentially "virulent nature." In Jean Baudrillard's 1981 treatise Simulacra and Simulation, the philosopher describes An American Family, arguably the first "reality" television series, as a marker of a new age in which the medium of television has a "viral, endemic, chronic, alarming presence."

Another formulation of the 'viral' concept includes the term media virus, or viral media, coined by Douglas Rushkoff, who defines it as a type of Trojan horse: "People are duped into passing a hidden agenda while circulating compelling content." Mosotho South-African media theorist Thomas Mofolo uses Rushkoff's idea to define viral as a type of virtual collective consciousness that primarily manifests via digital media networks and evolves into offline actions to produce a new social reality. Mofolo bases this definition on a study about how internet users involved in the Tunisian Arab Spring perceived the value of Facebook towards their revolution. Mofolo's understanding of the viral was first developed in a study on Global Citizen's #TogetherAtHome campaign and used to formulate a new theoretical framework called Hivemind Impact. Hivemind impact is a specific type of virality that is simulated via digital media networks with the goal of harnessing the virtual collective consciousness to take action on a social issue. For Mofolo, the viral eventually evolves into McLuhan's 'global village' when the virtual collective consciousness reaches a point of noogenesis that then becomes the noosphere.

Content sharing

See also: Content sharing

Early history

A page from Martin Luther's Ninety-five Theses, which was widely and rapidly distributed in 1517

Before writing and while most people were illiterate, the dominant means of spreading memes was oral culture like folk tales, folk songs, and oral poetry, which mutated over time as each retelling presented an opportunity for change. The printing press provided an easy way to copy written texts instead of handwritten manuscripts. In particular, pamphlets could be published in only a day or two, unlike books which took longer. For example, Martin Luther's Ninety-five Theses took only two months to spread throughout Europe. A study of United States newspapers in the 1800s found human-interest, "news you can use" stories and list-focused articles circulated nationally as local papers mailed copies to each other and selected content for reprinting. Chain letters spread by postal mail throughout the 1900s.

Urban legends also began as word-of-mouth memes. Like hoaxes, they are examples of falsehoods that people swallow, and, like them, often achieve broad public notoriety.

CompuServe

Beyond vocal sharing, the 20th century made huge strides in the World Wide Web and the ability to content share. In 1979, dial-up internet service provided by the company CompuServ was a key player in online communications and how information began spreading beyond the print. Those with access to a computer in the earliest of stages could not comprehend the full effect that public access to the internet could or would create. It is hard to remember the times of newspapers being delivered to households across the country in order to receive their news for the day, and it was when The Columbus Dispatch out of Columbus, Ohio broke barriers when it was first to publish in online format. The success that was predicted by CompuServe and the Associated Press led to some of the largest newspapers to become part of the movement to publish the news via online format. Content sharing in the journalism world brings new advances to viral aspects of how news is spread in a matter of seconds.

Internet memes

Main article: Internet meme

The creation of the Internet enabled users to select and share content with each other electronically, providing new, faster, and more decentralized controlled channels for spreading memes. Email forwards are essentially text memes, often including jokes, hoaxes, email scams, written versions of urban legends, political messages, and digital chain letters; if widely forwarded they might be called 'viral emails'. User-friendly consumer photo editing tools like Photoshop and image-editing websites have facilitated the creation of the genre of the image macro, where a popular image is overlaid with different humorous text phrases. These memes are typically created with Impact font. The growth of video-sharing websites like YouTube made viral videos possible.

It is sometimes difficult to predict which images and videos will "go viral"; sometimes the creation of a new Internet celebrity is a sudden surprise. One of the first documented viral videos is "Numa Numa", a webcam video of then-19-year-old Gary Brolsma lip-syncing and dancing to the Romanian pop song "Dragostea Din Tei".

The sharing of text, images, videos, or links to this content have been greatly facilitated by social media such as Facebook and Twitter. Other mimicry memes carried by Internet media include hashtags, language variations like intentional misspellings, and fads like planking. The popularity and widespread distribution of Internet memes have gotten the attention of advertisers, creating the field of viral marketing. A person, group, or company desiring much fast, cheap publicity might create a hashtag, image, or video designed to go viral; many such attempts are unsuccessful, but the few posts that "go viral" generate much publicity.

Types of viral phenomena

Viral videos

Main article: Viral video

Viral videos are among the most common type of viral phenomena. A viral video is any clip of animation or film that is spread rapidly through online sharing. Viral videos can receive millions of views as they are shared on social media sites, reposted to blogs, sent in emails and so on. When a video goes viral it has become very popular. Its exposure on the Internet grows exponentially as more and more people discover it and share it with others. An article or an image can also become viral.

The classification is probably assigned more as a result of intensive activity and the rate of growth among users in a relatively short amount of time than of simply how many hits something receives. Most viral videos contain humor and fall into broad categories:

• Unintentional: Videos that the creators never intended to go viral. These videos may have been posted by the creator or shared with friends, who then spread the content.
• Humorous: Videos that have been created specifically to entertain people. If a video is funny enough, it will spread.
• Promotional: Videos that are designed to go viral with a marketing message to raise brand awareness. Promotional viral videos fall under viral marketing practices. For instance, one of the newest viral commercial video – Extra Gum commercial.
• Charity: Videos created and spread in order to collect donations. For instance, Ice Bucket challenge was a hit on social networks in the summer of 2014.
• Art performances: a video created by artists to raise the problem, express ideas and the freedom of creativity.
• Political: Viral videos are powerful tools for politicians to boost their popularity. Barack Obama campaign launched Yes We Can slogan as a viral video on YouTube. "The Obama campaign posted almost 800 videos on YouTube, and the McCain campaign posted just over 100. The pro-Obama video "Yes we can" went viral after being uploaded to YouTube in February 2008." Other political viral videos served not as a promotion but as an agent for support and unification. Social media was actively employed in the Arab Spring. "The Tunisian uprising had special resonance in Egypt because it was prompted by incidents of police corruption and viral social media condemnation of them."

YouTube effect

With the creation of YouTube, a video-sharing website, there has been a huge surge in the number of viral videos on the Internet. This is primarily due to the ease of access to these videos and the ease of sharing them via social media websites. The ability to share videos from one person to another with ease means there are many cases of 'overnight' viral videos. "YouTube, which makes it easy to embed its content elsewhere have the freedom and mobility once ascribed to papyrus, enabling their rapid circulation across a range of social networks." YouTube has overtaken television in terms of the size of audience. As one example, American Idol was the most viewable TV show in 2009 in the U.S. while "a video of Scottish woman Susan Boyle auditioning for Britain's Got Talent with her singing was viewed more than 77 million times on YouTube". The capacity to attract an enormous audience on a user-friendly platform is one of the leading factors why YouTube generates viral videos. YouTube contributes to viral phenomenon spreadability since the idea of the platform is based on sharing and contribution. "Sites such as YouTube, eBay, Facebook, Flickr, Craigslist, and Wikipedia, only exist and have value because people use and contribute to them, and they are clearly better the more people are using and contributing to them. This is the essence of Web 2.0."

An example of one of the most prolific viral YouTube videos that fall into the promotional viral videos category is Kony 2012. On March 5, 2012, the charity organization Invisible Children Inc. posted a short film about the atrocities committed in Uganda by Joseph Kony and his rebel army. Artists use YouTube as their one of the main branding and communication platform to spread videos and make them viral. For instance, after her time off, Adele released her most-viewed song "Hello". "Hello" crossed 100 million views in just five days, making it the fastest video to reach it in 2015. YouTube viral videos make stars. As an example, Justin Bieber who was discovered since his video on YouTube Chris Brown's song "With You" went viral. Since its launch in 2005, YouTube has become a hub for aspiring singers and musicians. Talent managers look to it to find budding pop stars.

According to Visible Measures, the original "Kony 2012" video documentary, and the hundreds of excerpts and responses uploaded by audiences across the Web, collectively garnered 100 million views in a record six days. This example of how quickly the video spread emphasizes how YouTube acts as a catalyst in the spread of viral media. YouTube is considered as "multiple existing forms of participatory culture" and that trend is useful for the sake of business. "The discourse of Web 2.0 its power has been its erasure of this larger history of participatory practices, with companies acting as if they were "bestowing" agency onto audiences, making their creative output meaningful by valuing it within the logics of commodity culture."

Viral marketing

Main article: Viral marketing

Viral marketing is the phenomenon in which people actively assess media or content and decide to spread to others such as making a word-of-mouth recommendation, passing content through social media, posting a video to YouTube. The term was first popularized in 1995, after Hotmail spreading their service offer "Get your free web-base email at Hotmail." Viral marketing has become important in the business field in building brand recognition, with companies trying to get their customers and other audiences involved in circulating and sharing their content on social media both in voluntary and involuntary ways. Many brands undertake guerrilla marketing or buzz marketing to gain public attention. Some marketing campaigns seek to engage an audience to unwittingly pass along their campaign message.

The use of viral marketing is shifting from the concept that the content drives its own attention to the intended attempt to draw the attention. The companies are worried about making their content 'go viral' and how their customers' communication has the potential to circulate it widely. There has been much discussion about morality in doing viral marketing. Iain Short (2010) points out that many applications on Twitter and Facebook generates automated marketing message and update it on the audience's personal timelines without users personally pass it along.

Stacy Wood from North Carolina State University has conducted research and found that the value of recommendations from 'everyday people' has a potential impact on the brands. Consumers have been bombarded by thousands of messages every day which makes authenticity and credibility of marketing message been questioned; word of mouth from 'everyday people' therefore becomes an incredibly important source of credible information. If a company sees that the word-of-mouth from "the average person" is crucial for the greater opportunity for influencing others, many questions remain. "What implicit contracts exist between brands and those recommenders? What moral codes and guidelines should brands respect when encouraging, soliciting, or reacting to comments from those audiences they wish to reach? What types of compensation, if any, do audience members deserve for their promotional labor when they provide a testimonial."

An example of effective viral marketing can be the unprecedented boost in sales of the Popeyes chicken sandwich. After the Twitter account for Chick-fil-A attempted to undercut Popeyes by suggesting that Popeyes' chicken sandwich wasn't the "original chicken sandwich", Popeyes responded with a tweet that would end up going viral. After the response had amassed 85,000 retweets and 300,000 likes, Popeyes chains began to sell many more sandwiches to the point where many locations sold all of their stock of chicken sandwiches. This prompted other chicken chains to tweet about their chicken sandwiches, but none of these efforts became as widespread as it was for Popeyes.

Financial contagion

Main article: Financial contagion

In macroeconomics, "financial contagion" is a proposed socially-viral phenomenon wherein disturbances quickly spread across global financial markets.

Evaluation by commentators

Some social commentators have a negative view of "viral" content, though others are neutral or celebrate the democratization of content as compared to the gatekeepers of older media. According to the authors of Spreadable Media: Creating Value and Meaning in a Networked Culture: "Ideas are transmitted, often without critical assessment, across a broad array of minds and this uncoordinated flow of information is associated with "bad ideas" or "ruinous fads and foolish fashions." Science fiction sometimes discusses 'viral' content "describing (generally bad) ideas that spread like germs." For example, the 1992 novel Snow Crash explores the implications of an ancient memetic meta-virus and its modern-day computer virus equivalent:

We are all susceptible to the pull of viral ideas. Like mass hysteria. Or a tune that gets into your head that you keep on humming all day until you spread it to someone else. Jokes. Urban legends. Crackpot religions. No matter how smart we get, there is always this deep irrational part that makes us potential hosts for self-replicating information.

— Snow Crash (1992)

The spread of viral phenomena is also regarded as part of the cultural politics of network culture or the virality of the age of networks. Network culture enables the audience to create and spread viral content. "Audiences play an active role in 'spreading' content rather than serving as passive carriers of viral media: their choices, investments, agendas, and actions determine what gets valued." Various authors have pointed to the intensification in connectivity brought about by network technologies as a possible trigger for increased chances of infection from wide-ranging social, cultural, political, and economic contagions. For example, the social scientist Jan van Dijk warns of new vulnerabilities that arise when network society encounters "too much connectivity." The proliferation of global transport networks makes this model of society susceptible to the spreading of biological diseases. Digital networks become volatile under the destructive potential of computer viruses and worms. Enhanced by the rapidity and extensiveness of technological networks, the spread of social conformity, political rumor, fads, fashions, gossip, and hype threatens to destabilize established political order.

Links between viral phenomena that spread on digital networks and the early sociological theories of Gabriel Tarde have been made in digital media theory by Tony D Sampson (2012; 2016). In this context, Tarde's social imitation thesis is used to argue against the biological deterministic theories of cultural contagion forwarded in memetics. In its place, Sampson proposes a Tarde-inspired somnambulist media theory of the viral.

Molecular Hamiltonian

From Wikipedia, the free encyclopedia

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

In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.

The elementary parts of a molecule are the nuclei, characterized by their atomic numbers, Z, and the electrons, which have negative elementary charge, −e. Their interaction gives a nuclear charge of Z + q, where q = −eN, with N equal to the number of electrons. Electrons and nuclei are, to a very good approximation, point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the kinetic energies of the electrons and the Coulomb (electrostatic) interactions between the two kinds of charged particles. The Hamiltonian that contains only the kinetic energies of electrons and nuclei, and the Coulomb interactions between them, is known as the Coulomb Hamiltonian. From it are missing a number of small terms, most of which are due to electronic and nuclear spin.

Although it is generally assumed that the solution of the time-independent Schrödinger equation associated with the Coulomb Hamiltonian will predict most properties of the molecule, including its shape (three-dimensional structure), calculations based on the full Coulomb Hamiltonian are very rare. The main reason is that its Schrödinger equation is very difficult to solve. Applications are restricted to small systems like the hydrogen molecule.

Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised by Born and Oppenheimer. The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons only. The stationary nuclei enter the problem only as generators of an electric potential in which the electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so-called clamped nucleus Hamiltonian, also called electronic Hamiltonian, that acts only on functions of the electronic coordinates.

Once the Schrödinger equation of the clamped nucleus Hamiltonian has been solved for a sufficient number of constellations of the nuclei, an appropriate eigenvalue (usually the lowest) can be seen as a function of the nuclear coordinates, which leads to a potential energy surface. In practical calculations the surface is usually fitted in terms of some analytic functions. In the second step of the Born–Oppenheimer approximation the part of the full Coulomb Hamiltonian that depends on the electrons is replaced by the potential energy surface. This converts the total molecular Hamiltonian into another Hamiltonian that acts only on the nuclear coordinates. In the case of a breakdown of the Born–Oppenheimer approximation—which occurs when energies of different electronic states are close—the neighboring potential energy surfaces are needed, see this article for more details on this.

The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory) frame, but then the translational and rotational (external) energies are not accounted for. Only the (internal) atomic vibrations enter the problem. Further, for molecules larger than triatomic ones, it is quite common to introduce the harmonic approximation, which approximates the potential energy surface as a quadratic function of the atomic displacements. This gives the harmonic nuclear motion Hamiltonian. Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one-dimensional harmonic oscillator Hamiltonians. The one-dimensional harmonic oscillator is one of the few systems that allows an exact solution of the Schrödinger equation.

Alternatively, the nuclear motion (rovibrational) Schrödinger equation can be solved in a special frame (an Eckart frame) that rotates and translates with the molecule. Formulated with respect to this body-fixed frame the Hamiltonian accounts for rotation, translation and vibration of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it is often referred to as Watson's nuclear motion Hamiltonian, but it is also known as the Eckart Hamiltonian.

Coulomb Hamiltonian

The algebraic form of many observables—i.e., Hermitian operators representing observable quantities—is obtained by the following quantization rules:

• Write the classical form of the observable in Hamilton form (as a function of momenta p and positions q). Both vectors are expressed with respect to an arbitrary inertial frame, usually referred to as laboratory-frame or space-fixed frame.
• Replace p by ${\displaystyle -i\hbar {\boldsymbol {\nabla }}}$ and interpret q as a multiplicative operator. Here ${\displaystyle {\boldsymbol {\nabla }}}$ is the nabla operator, a vector operator consisting of first derivatives. The well-known commutation relations for the p and q operators follow directly from the differentiation rules.

Classically the electrons and nuclei in a molecule have kinetic energy of the form p²/(2 m) and interact via Coulomb interactions, which are inversely proportional to the distance r_ij between particle i and j.

$${\displaystyle r_{ij}\equiv |\mathbf {r} _{i}-\mathbf {r} _{j}|={\sqrt {(\mathbf {r} _{i}-\mathbf {r} _{j})\cdot (\mathbf {r} _{i}-\mathbf {r} _{j})}}={\sqrt {(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}}}.}$$

In this expression r_i stands for the coordinate vector of any particle (electron or nucleus), but from here on we will reserve capital R to represent the nuclear coordinate, and lower case r for the electrons of the system. The coordinates can be taken to be expressed with respect to any Cartesian frame centered anywhere in space, because distance, being an inner product, is invariant under rotation of the frame and, being the norm of a difference vector, distance is invariant under translation of the frame as well.

By quantizing the classical energy in Hamilton form one obtains the a molecular Hamilton operator that is often referred to as the Coulomb Hamiltonian. This Hamiltonian is a sum of five terms. They are

1. The kinetic energy operators for each nucleus in the system;
   $${\displaystyle {\hat {T}}_{n}=-\sum _{i}{\frac {\hbar ^{2}}{2M_{i}}}\nabla _{\mathbf {R} _{i}}^{2}}$$

   
2. The kinetic energy operators for each electron in the system;
   $${\displaystyle {\hat {T}}_{e}=-\sum _{i}{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{\mathbf {r} _{i}}^{2}}$$

   
3. The potential energy between the electrons and nuclei – the total electron-nucleus Coulombic attraction in the system;
   $${\displaystyle {\hat {U}}_{en}=-\sum _{i}\sum _{j}{\frac {Z_{i}e^{2}}{4\pi \varepsilon _{0}\left|\mathbf {R} _{i}-\mathbf {r} _{j}\right|}}}$$

   
4. The potential energy arising from Coulombic electron-electron repulsions
   $${\displaystyle {\hat {U}}_{ee}={1 \over 2}\sum _{i}\sum _{j\neq i}{\frac {e^{2}}{4\pi \varepsilon _{0}\left|\mathbf {r} _{i}-\mathbf {r} _{j}\right|}}=\sum _{i}\sum _{j>i}{\frac {e^{2}}{4\pi \varepsilon _{0}\left|\mathbf {r} _{i}-\mathbf {r} _{j}\right|}}}$$

   
5. The potential energy arising from Coulombic nuclei-nuclei repulsions – also known as the nuclear repulsion energy. See electric potential for more details.
   $${\displaystyle {\hat {U}}_{nn}={1 \over 2}\sum _{i}\sum _{j\neq i}{\frac {Z_{i}Z_{j}e^{2}}{4\pi \varepsilon _{0}\left|\mathbf {R} _{i}-\mathbf {R} _{j}\right|}}=\sum _{i}\sum _{j>i}{\frac {Z_{i}Z_{j}e^{2}}{4\pi \varepsilon _{0}\left|\mathbf {R} _{i}-\mathbf {R} _{j}\right|}}.}$$

   

Here M_i is the mass of nucleus i, Z_i is the atomic number of nucleus i, and m_e is the mass of the electron. The Laplace operator of particle i is:${\displaystyle \nabla _{\mathbf {r} _{i}}^{2}\equiv {\boldsymbol {\nabla }}_{\mathbf {r} _{i}}\cdot {\boldsymbol {\nabla }}_{\mathbf {r} _{i}}={\frac {\partial ^{2}}{\partial x_{i}^{2}}}+{\frac {\partial ^{2}}{\partial y_{i}^{2}}}+{\frac {\partial ^{2}}{\partial z_{i}^{2}}}}$. Since the kinetic energy operator is an inner product, it is invariant under rotation of the Cartesian frame with respect to which x_i, y_i, and z_i are expressed.

Small terms

In the 1920s much spectroscopic evidence made it clear that the Coulomb Hamiltonian is missing certain terms. Especially for molecules containing heavier atoms, these terms, although much smaller than kinetic and Coulomb energies, are nonnegligible. These spectroscopic observations led to the introduction of a new degree of freedom for electrons and nuclei, namely spin. This empirical concept was given a theoretical basis by Paul Dirac when he introduced a relativistically correct (Lorentz covariant) form of the one-particle Schrödinger equation. The Dirac equation predicts that spin and spatial motion of a particle interact via spin–orbit coupling. In analogy spin-other-orbit coupling was introduced. The fact that particle spin has some of the characteristics of a magnetic dipole led to spin–spin coupling. Further terms without a classical counterpart are the Fermi-contact term (interaction of electronic density on a finite size nucleus with the nucleus), and nuclear quadrupole coupling (interaction of a nuclear quadrupole with the gradient of an electric field due to the electrons). Finally a parity violating term predicted by the Standard Model must be mentioned. Although it is an extremely small interaction, it has attracted a fair amount of attention in the scientific literature because it gives different energies for the enantiomers in chiral molecules.

The remaining part of this article will ignore spin terms and consider the solution of the eigenvalue (time-independent Schrödinger) equation of the Coulomb Hamiltonian.

The Schrödinger equation of the Coulomb Hamiltonian

The Coulomb Hamiltonian has a continuous spectrum due to the center of mass (COM) motion of the molecule in homogeneous space. In classical mechanics it is easy to separate off the COM motion of a system of point masses. Classically the motion of the COM is uncoupled from the other motions. The COM moves uniformly (i.e., with constant velocity) through space as if it were a point particle with mass equal to the sum M_tot of the masses of all the particles.

In quantum mechanics a free particle has as state function a plane wave function, which is a non-square-integrable function of well-defined momentum. The kinetic energy of this particle can take any positive value. The position of the COM is uniformly probable everywhere, in agreement with the Heisenberg uncertainty principle.

By introducing the coordinate vector X of the center of mass as three of the degrees of freedom of the system and eliminating the coordinate vector of one (arbitrary) particle, so that the number of degrees of freedom stays the same, one obtains by a linear transformation a new set of coordinates t_i. These coordinates are linear combinations of the old coordinates of all particles (nuclei and electrons). By applying the chain rule one can show that

$${\displaystyle H=-{\frac {\hbar ^{2}}{2M_{\textrm {tot}}}}\nabla _{\mathbf {X} }^{2}+H'\quad {\text{with }}\quad H'=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N_{\textrm {tot}}-1}{\frac {1}{m_{i}}}\nabla _{i}^{2}+{\frac {\hbar ^{2}}{2M_{\textrm {tot}}}}\sum _{i,j=1}^{N_{\textrm {tot}}-1}\nabla _{i}\cdot \nabla _{j}+V(\mathbf {t} ).}$$

The first term of ${\displaystyle H}$ is the kinetic energy of the COM motion, which can be treated separately since ${\displaystyle H'}$ does not depend on X. As just stated, its eigenstates are plane waves. The potential V(t) consists of the Coulomb terms expressed in the new coordinates. The first term of ${\displaystyle H'}$ has the usual appearance of a kinetic energy operator. The second term is known as the mass polarization term. The translationally invariant Hamiltonian ${\displaystyle H'}$ can be shown to be self-adjoint and to be bounded from below. That is, its lowest eigenvalue is real and finite. Although ${\displaystyle H'}$ is necessarily invariant under permutations of identical particles (since ${\displaystyle H}$ and the COM kinetic energy are invariant), its invariance is not manifest.

Not many actual molecular applications of ${\displaystyle H'}$ exist; see, however, the seminal work on the hydrogen molecule for an early application. In the great majority of computations of molecular wavefunctions the electronic problem is solved with the clamped nucleus Hamiltonian arising in the first step of the Born–Oppenheimer approximation.

Clamped nucleus Hamiltonian

The clamped nucleus Hamiltonian describes the energy of the electrons in the electrostatic field of the nuclei, where the nuclei are assumed to be stationary with respect to an inertial frame. The form of the electronic Hamiltonian is

$${\displaystyle {\hat {H}}_{\mathrm {el} }={\hat {T}}_{e}+{\hat {U}}_{en}+{\hat {U}}_{ee}+{\hat {U}}_{nn}.}$$

The coordinates of electrons and nuclei are expressed with respect to a frame that moves with the nuclei, so that the nuclei are at rest with respect to this frame. The frame stays parallel to a space-fixed frame. It is an inertial frame because the nuclei are assumed not to be accelerated by external forces or torques. The origin of the frame is arbitrary, it is usually positioned on a central nucleus or in the nuclear center of mass. Sometimes it is stated that the nuclei are "at rest in a space-fixed frame". This statement implies that the nuclei are viewed as classical particles, because a quantum mechanical particle cannot be at rest. (It would mean that it had simultaneously zero momentum and well-defined position, which contradicts Heisenberg's uncertainty principle).

Since the nuclear positions are constants, the electronic kinetic energy operator is invariant under translation over any nuclear vector. The Coulomb potential, depending on difference vectors, is invariant as well. In the description of atomic orbitals and the computation of integrals over atomic orbitals this invariance is used by equipping all atoms in the molecule with their own localized frames parallel to the space-fixed frame.

As explained in the article on the Born–Oppenheimer approximation, a sufficient number of solutions of the Schrödinger equation of ${\displaystyle H_{\text{el}}}$ leads to a potential energy surface (PES) ${\displaystyle V(\mathbf {R} _{1},\mathbf {R} _{2},\ldots ,\mathbf {R} _{N})}$. It is assumed that the functional dependence of V on its coordinates is such that

$${\displaystyle V(\mathbf {R} _{1},\mathbf {R} _{2},\ldots ,\mathbf {R} _{N})=V(\mathbf {R} '_{1},\mathbf {R} '_{2},\ldots ,\mathbf {R} '_{N})}$$

for

$${\displaystyle \mathbf {R} '_{i}=\mathbf {R} _{i}+\mathbf {t} \;\;{\text{(translation) and}}\;\;\mathbf {R} '_{i}=\mathbf {R} _{i}+{\frac {\Delta \phi }{|\mathbf {s} |}}\;(\mathbf {s} \times \mathbf {R} _{i})\;\;{\text{(infinitesimal rotation)}},}$$

where t and s are arbitrary vectors and Δφ is an infinitesimal angle, Δφ >> Δφ². This invariance condition on the PES is automatically fulfilled when the PES is expressed in terms of differences of, and angles between, the R_i, which is usually the case.

Harmonic nuclear motion Hamiltonian

In the remaining part of this article we assume that the molecule is semi-rigid. In the second step of the BO approximation the nuclear kinetic energy T_n is reintroduced and the Schrödinger equation with Hamiltonian

$${\displaystyle {\hat {H}}_{\mathrm {nuc} }=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}\sum _{\alpha =1}^{3}{\frac {1}{M_{i}}}{\frac {\partial ^{2}}{\partial R_{i\alpha }^{2}}}+V(\mathbf {R} _{1},\ldots ,\mathbf {R} _{N})}$$

is considered. One would like to recognize in its solution: the motion of the nuclear center of mass (3 degrees of freedom), the overall rotation of the molecule (3 degrees of freedom), and the nuclear vibrations. In general, this is not possible with the given nuclear kinetic energy, because it does not separate explicitly the 6 external degrees of freedom (overall translation and rotation) from the 3N − 6 internal degrees of freedom. In fact, the kinetic energy operator here is defined with respect to a space-fixed (SF) frame. If we were to move the origin of the SF frame to the nuclear center of mass, then, by application of the chain rule, nuclear mass polarization terms would appear. It is customary to ignore these terms altogether and we will follow this custom.

In order to achieve a separation we must distinguish internal and external coordinates, to which end Eckart introduced conditions to be satisfied by the coordinates. We will show how these conditions arise in a natural way from a harmonic analysis in mass-weighted Cartesian coordinates.

In order to simplify the expression for the kinetic energy we introduce mass-weighted displacement coordinates

$${\displaystyle {\boldsymbol {\rho }}_{i}\equiv {\sqrt {M_{i}}}(\mathbf {R} _{i}-\mathbf {R} _{i}^{0})}$$

. Since

$${\displaystyle {\frac {\partial }{\partial \rho _{i\alpha }}}={\frac {\partial }{{\sqrt {M_{i}}}(\partial R_{i\alpha }-\partial R_{i\alpha }^{0})}}={\frac {1}{\sqrt {M_{i}}}}{\frac {\partial }{\partial R_{i\alpha }}},}$$

the kinetic energy operator becomes,

$${\displaystyle T=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}\sum _{\alpha =1}^{3}{\frac {\partial ^{2}}{\partial \rho _{i\alpha }^{2}}}.}$$

If we make a Taylor expansion of V around the equilibrium geometry,

$${\displaystyle V=V_{0}+\sum _{i=1}^{N}\sum _{\alpha =1}^{3}{\Big (}{\frac {\partial V}{\partial \rho _{i\alpha }}}{\Big )}_{0}\;\rho _{i\alpha }+{\frac {1}{2}}\sum _{i,j=1}^{N}\sum _{\alpha ,\beta =1}^{3}{\Big (}{\frac {\partial ^{2}V}{\partial \rho _{i\alpha }\partial \rho _{j\beta }}}{\Big )}_{0}\;\rho _{i\alpha }\rho _{j\beta }+\cdots ,}$$

and truncate after three terms (the so-called harmonic approximation), we can describe V with only the third term. The term V₀ can be absorbed in the energy (gives a new zero of energy). The second term is vanishing because of the equilibrium condition. The remaining term contains the Hessian matrix F of V, which is symmetric and may be diagonalized with an orthogonal 3N × 3N matrix with constant elements:

$${\displaystyle \mathbf {Q} \mathbf {F} \mathbf {Q} ^{\mathrm {T} }={\boldsymbol {\Phi }}\quad {\text{with}}\quad {\boldsymbol {\Phi }}=\operatorname {diag} (f_{1},\dots ,f_{3N-6},0,\ldots ,0).}$$

It can be shown from the invariance of V under rotation and translation that six of the eigenvectors of F (last six rows of Q) have eigenvalue zero (are zero-frequency modes). They span the external space. The first 3N − 6 rows of Q are—for molecules in their ground state—eigenvectors with non-zero eigenvalue; they are the internal coordinates and form an orthonormal basis for a (3N - 6)-dimensional subspace of the nuclear configuration space R^3N, the internal space. The zero-frequency eigenvectors are orthogonal to the eigenvectors of non-zero frequency. It can be shown that these orthogonalities are in fact the Eckart conditions. The kinetic energy expressed in the internal coordinates is the internal (vibrational) kinetic energy.

With the introduction of normal coordinates

$${\displaystyle q_{t}\equiv \sum _{i=1}^{N}\sum _{\alpha =1}^{3}\;Q_{t,i\alpha }\rho _{i\alpha },}$$

the vibrational (internal) part of the Hamiltonian for the nuclear motion becomes in the harmonic approximation

$${\displaystyle {\hat {H}}_{\text{nuc}}\approx {\frac {1}{2}}\sum _{t=1}^{3N-6}\left[-\hbar ^{2}{\frac {\partial ^{2}}{\partial q_{t}^{2}}}+f_{t}q_{t}^{2}\right].}$$

The corresponding Schrödinger equation is easily solved, it factorizes into 3N − 6 equations for one-dimensional harmonic oscillators. The main effort in this approximate solution of the nuclear motion Schrödinger equation is the computation of the Hessian F of V and its diagonalization.

This approximation to the nuclear motion problem, described in 3N mass-weighted Cartesian coordinates, became standard in quantum chemistry, since the days (1980s-1990s) that algorithms for accurate computations of the Hessian F became available. Apart from the harmonic approximation, it has as a further deficiency that the external (rotational and translational) motions of the molecule are not accounted for. They are accounted for in a rovibrational Hamiltonian that sometimes is called Watson's Hamiltonian.

Watson's nuclear motion Hamiltonian

In order to obtain a Hamiltonian for external (translation and rotation) motions coupled to the internal (vibrational) motions, it is common to return at this point to classical mechanics and to formulate the classical kinetic energy corresponding to these motions of the nuclei. Classically it is easy to separate the translational—center of mass—motion from the other motions. However, the separation of the rotational from the vibrational motion is more difficult and is not completely possible. This ro-vibrational separation was first achieved by Eckart in 1935 by imposing by what is now known as Eckart conditions. Since the problem is described in a frame (an "Eckart" frame) that rotates with the molecule, and hence is a non-inertial frame, energies associated with the fictitious forces: centrifugal and Coriolis force appear in the kinetic energy.

In general, the classical kinetic energy T defines the metric tensor g = (g_ij) associated with the curvilinear coordinates s = (s_i) through

$${\displaystyle 2T=\sum _{ij}g_{ij}{\dot {s}}_{i}{\dot {s}}_{j}.}$$

The quantization step is the transformation of this classical kinetic energy into a quantum mechanical operator. It is common to follow Podolsky by writing down the Laplace–Beltrami operator in the same (generalized, curvilinear) coordinates s as used for the classical form. The equation for this operator requires the inverse of the metric tensor g and its determinant. Multiplication of the Laplace–Beltrami operator by ${\displaystyle -\hbar ^{2}}$ gives the required quantum mechanical kinetic energy operator. When we apply this recipe to Cartesian coordinates, which have unit metric, the same kinetic energy is obtained as by application of the quantization rules.

The nuclear motion Hamiltonian was obtained by Wilson and Howard in 1936, who followed this procedure, and further refined by Darling and Dennison in 1940. It remained the standard until 1968, when Watson was able to simplify it drastically by commuting through the derivatives the determinant of the metric tensor. We will give the ro-vibrational Hamiltonian obtained by Watson, which often is referred to as the Watson Hamiltonian. Before we do this we must mention that a derivation of this Hamiltonian is also possible by starting from the Laplace operator in Cartesian form, application of coordinate transformations, and use of the chain rule. The Watson Hamiltonian, describing all motions of the N nuclei, is

$${\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2M_{\mathrm {tot} }}}\sum _{\alpha =1}^{3}{\frac {\partial ^{2}}{\partial X_{\alpha }^{2}}}+{\frac {1}{2}}\sum _{\alpha ,\beta =1}^{3}\mu _{\alpha \beta }({\mathcal {P}}_{\alpha }-\Pi _{\alpha })({\mathcal {P}}_{\beta }-\Pi _{\beta })+U-{\frac {\hbar ^{2}}{2}}\sum _{s=1}^{3N-6}{\frac {\partial ^{2}}{\partial q_{s}^{2}}}+V.}$$

The first term is the center of mass term

$${\displaystyle \mathbf {X} \equiv {\frac {1}{M_{\mathrm {tot} }}}\sum _{i=1}^{N}M_{i}\mathbf {R} _{i}\quad \mathrm {with} \quad M_{\mathrm {tot} }\equiv \sum _{i=1}^{N}M_{i}.}$$

The second term is the rotational term akin to the kinetic energy of the rigid rotor. Here ${\displaystyle {\mathcal {P}}_{\alpha }}$ is the α component of the body-fixed rigid rotor angular momentum operator, see this article for its expression in terms of Euler angles. The operator ${\displaystyle \Pi _{\alpha }\,}$ is a component of an operator known as the vibrational angular momentum operator (although it does not satisfy angular momentum commutation relations),

$${\displaystyle \Pi _{\alpha }=-i\hbar \sum _{s,t=1}^{3N-6}\zeta _{st}^{\alpha }\;q_{s}{\frac {\partial }{\partial q_{t}}}}$$

with the Coriolis coupling constant:

$${\displaystyle \zeta _{st}^{\alpha }=\sum _{i=1}^{N}\sum _{\beta ,\gamma =1}^{3}\epsilon _{\alpha \beta \gamma }Q_{s,i\beta }\,Q_{t,i\gamma }\;\;\mathrm {and} \quad \alpha =1,2,3.}$$

Here ε_αβγ is the Levi-Civita symbol. The terms quadratic in the ${\displaystyle {\mathcal {P}}_{\alpha }}$ are centrifugal terms, those bilinear in ${\displaystyle {\mathcal {P}}_{\alpha }}$ and ${\displaystyle \Pi _{\beta }\,}$ are Coriolis terms. The quantities Q _{s, iγ} are the components of the normal coordinates introduced above. Alternatively, normal coordinates may be obtained by application of Wilson's GF method. The 3 × 3 symmetric matrix ${\displaystyle {\boldsymbol {\mu }}}$ is called the effective reciprocal inertia tensor. If all q _s were zero (rigid molecule) the Eckart frame would coincide with a principal axes frame (see rigid rotor) and ${\displaystyle {\boldsymbol {\mu }}}$ would be diagonal, with the equilibrium reciprocal moments of inertia on the diagonal. If all q _s would be zero, only the kinetic energies of translation and rigid rotation would survive.

The potential-like term U is the Watson term:

$${\displaystyle U=-{\frac {1}{8}}\sum _{\alpha =1}^{3}\mu _{\alpha \alpha }}$$

proportional to the trace of the effective reciprocal inertia tensor.

The fourth term in the Watson Hamiltonian is the kinetic energy associated with the vibrations of the atoms (nuclei) expressed in normal coordinates q_s, which as stated above, are given in terms of nuclear displacements ρ_iα by

$${\displaystyle q_{s}=\sum _{i=1}^{N}\sum _{\alpha =1}^{3}Q_{s,i\alpha }\rho _{i\alpha }\quad {\text{for}}\quad s=1,\ldots ,3N-6.}$$

Finally V is the unexpanded potential energy by definition depending on internal coordinates only. In the harmonic approximation it takes the form

$${\displaystyle V\approx {\frac {1}{2}}\sum _{s=1}^{3N-6}f_{s}q_{s}^{2}.}$$