Hückel method

From Wikipedia, the free encyclopedia

The Hückel method or Hückel molecular orbital method (HMO), proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals method for the determination of energies of molecular orbitals of π electrons in conjugated hydrocarbon systems, such as ethylene, benzene and butadiene.^[1][2] It is the theoretical basis for Hückel's rule for the aromaticity of (4n + 2) π electron cyclic, planar systems. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon, known in this context as heteroatoms.^[3] A more dramatic extension of the method to include σ electrons, known as the extended Hückel method, was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general (not just planar systems) and was used to test the Woodward–Hoffmann rules.^[4]

In spite of its simplicity, the Hückel method in its original form makes qualitatively accurate and chemically useful predictions for many common molecules and is therefore a powerful educational tool. It is described in many introductory quantum chemistry and physical organic chemistry textbooks.

Hückel characteristics

The method has several characteristics:

• It limits itself to conjugated hydrocarbons.
• Only π electron molecular orbitals are included because these determine much of the chemical and spectral properties of these molecules. The σ electrons are assumed to form the framework of the molecule and σ connectivity is used to determine whether two π orbitals interact. However, the orbitals formed by σ electrons are ignored and assumed not to interact with π electrons. This is referred to as σ-π separability. It is justified by the orthogonality of σ and π orbitals in planar molecules. For this reason, the Hückel method is limited to systems that are planar or nearly so.
• The method is based on applying the variational method to linear combination of atomic orbitals and making simplifying assumptions regarding the overlap, resonance and Coulomb integrals of these atomic orbitals. It does not attempt to solve the Schrödinger equation, and neither the functional form of the basis atomic orbitals nor details of the Hamiltonian are involved.
• Interestingly, for hydrocarbons, the method takes atomic connectivity as the only input; empirical parameters are only needed when heteroatoms are introduced.
• The method predicts how many energy levels exist for a given molecule, which levels are degenerate and it expresses the molecular orbital energies in terms of two parameters, called α, the energy of an electron in a 2p orbital, and β, the interaction energy between two 2p orbitals (the extent to which an electron is stabilized by allowing it to delocalize between two orbitals). To understand and compare systems in a qualitative or even semi-quantitative sense, explicit numerical values for these parameters are typically not required.
• In addition the method also enables calculation of charge density for each atom in the π framework, the fractional bond order between any two atoms, and the overall molecular dipole moment.

Hückel results

The results for a few simple molecules are tabulated below:

Molecule	Energy	Frontier orbital	HOMO–LUMO energy gap
Ethylene	E1 = α – β	LUMO	−2β
	E2 = α + β	HOMO
Butadiene	E1 = α + 1.618...β	1.618... and 0.618... are ${\displaystyle ({\sqrt {5}}\pm 1)/2}$
	E2 = α + 0.618...β	HOMO	–1.236...β
	E3 = α – 0.618...β	LUMO
	E4 = α – 1.618...β
Benzene	E1 = α + 2β
	E2 = α + β
	E3 = α + β	HOMO	−2β
	E4 = α − β	LUMO
	E5 = α − β
	E6 = α − 2β
Cyclobutadiene	E1 = α + 2β
	E2 = α	SOMO	0
	E3 = α	SOMO
	E4 = α − 2β
Table 1. Hückel method results.  Lowest energies of top α and β are both negative values.[5] HOMO/LUMO/SOMO = Highest occupied/lowest unoccupied/singly-occupied molecular orbitals.

The theory predicts two energy levels for ethylene with its two π electrons filling the low-energy HOMO and the high energy LUMO remaining empty. In butadiene the 4 π-electrons occupy 2 low energy molecular orbitals, out of a total of 4, and for benzene 6 energy levels are predicted, two of them degenerate.

For linear and cyclic systems (with N atoms), general solutions exist:^[6]

Frost circle (de) mnemonic for 1,3-cyclopenta-5-dienyl anion

• Linear system (polyene/polyenyl): ${\displaystyle E_{k}=\alpha +2\beta \cos {\frac {(k+1)\pi }{N+1}}\quad (k=0,1,\ldots ,N-1)}$.
  • Energy levels are all distinct.

• Cyclic system, Hückel topology (annulene/annulenyl): ${\displaystyle E_{k}=\alpha +2\beta \cos {\frac {2k\pi }{N}}\quad (k=0,1,\ldots ,\lfloor N/2\rfloor )}$.
  • Energy levels ${\displaystyle k=1,\ldots ,\lceil N/2\rceil -1}$ are each doubly degenerate.
• Cyclic system, Möbius topology (hypothetical^[7]): ${\displaystyle E_{k}=\alpha +2\beta '\cos {\frac {(2k+1)\pi }{N}},\ \beta '=\beta \cos(\pi /N)\quad (k=0,1,\ldots ,\lceil N/2\rceil -1)}$.
  • Energy levels ${\displaystyle k=0,\ldots ,\lfloor N/2\rfloor -1}$ are each doubly degenerate.

The energy levels for cyclic systems can be predicted using the Frost circle (de) mnemonic (named after the American chemist Arthur Atwater Frost (de)). A circle centered at α with radius 2β is inscribed with a polygon with one vertex pointing down; the vertices represent energy levels with the appropriate energies.^[8] A related mnemonic exists for linear and Möbius systems.^[9]

The value for β in Hückel theory is roughly constant for structurally similar compounds, but not surprisingly, structurally dissimilar compounds will give very different values for β. For example, using the π bond energy of ethylene (65 kcal/mole) and comparing the energy of a doubly-occupied π orbital (2α + 2β) with the energy of two isolated p orbitals (2α), a value of β = 32.5 kcal/mole can be inferred. On the other hand, using the resonance energy of benzene (36 kcal/mole, derived from heats of hydrogenation) and comparing benzene (6α + 8β) with a hypothetical "non-aromatic 1,3,5-cyclohexatriene" (6α + 6β), a much smaller value of β = 18 kcal/mole emerges. These differences are not surprising, given the substantially shorter bond length of ethylene (1.33 Å) compared to benzene (1.40 Å). The shorter distance between the interacting p orbitals accounts for the greater energy of interaction, which is reflected by a higher value of β. Nevertheless, Hückel theory should not be expected to provide accurate quantitative predictions; only semi-quantitative and qualitative trends and patterns are reliable.

With this caveat in mind, many predictions of the theory have been experimentally verified:

• The HOMO–LUMO gap, in terms of the β constant, correlates directly with the respective molecular electronic transitions observed with UV/VIS spectroscopy. For linear polyenes, the energy gap is given as:

${\displaystyle \Delta E=-4\beta \sin {\frac {\pi }{2(n+1)}}}$

from which a value for β can be obtained between −60 and −70 kcal/mol (−250 to −290 kJ/mol).^[10]

• The predicted molecular orbital energies as stipulated by Koopmans' theorem correlate with photoelectron spectroscopy.^[11]
• The Hückel delocalization energy correlates with the experimental heat of combustion. This energy is defined as the difference between the total predicted π energy (in benzene 8β) and a hypothetical π energy in which all ethylene units are assumed isolated, each contributing 2β (making benzene 3 × 2β = 6β).
• Molecules with molecular orbitals paired up such that only the sign differs (for example α ± β) are called alternant hydrocarbons and have in common small molecular dipole moments. This is in contrast to non-alternant hydrocarbons, such as azulene and fulvene that have large dipole moments. The Hückel theory is more accurate for alternant hydrocarbons.
• For cyclobutadiene the theory predicts that the two high-energy electrons occupy a degenerate pair of molecular orbitals that are neither stabilized or destabilized. Hence the square molecule would be a very reactive triplet diradical (the ground state is actually rectangular without degenerate orbitals). In fact, all cyclic conjugated hydrocarbons with a total of 4nπ electrons share this molecular orbital pattern, and this forms the basis of Hückel's rule.
• Dewar reactivity numbers deriving from the Hückel approach correctly predict the reactivity of aromatic systems with nucleophiles and electrophiles.

Mathematics behind the Hückel method

The mathematics of the Hückel method is based on the Ritz method. In short, given a basis set of n normalized atomic orbitals ${\displaystyle \{\phi _{i}\}_{i=1}^{n}}$, an ansatz molecular orbital ${\displaystyle \psi _{g}=N(c_{1}\phi _{1}+\cdots +c_{n}\phi _{n})}$ is written down, with normalization constant N and coefficients ${\displaystyle c_{i}}$ which are to be determined. In other words, we are assuming that the molecular orbital (MO) can be written as a linear combination of atomic orbitals, a conceptually intuitive and convenient approximation (the linear combination of atomic orbitals or LCAO approximation). The variational theorem states that given an eigenvalue problem ${\displaystyle {\hat {H}}\psi ^{(i)}=E^{(i)}\psi ^{(i)}}$ with smallest eigenvalue ${\displaystyle E^{(0)}}$ and corresponding wavefunction ${\displaystyle \psi ^{(0)}}$, any normalized trial wavefunction ${\displaystyle \psi _{g}}$ (i.e., ${\textstyle \int _{\mathbb {R} ^{3}}\psi _{g}^{*}\,\psi _{g}\,dV=1}$ holds) will satisfy

${\displaystyle {\mathcal {E}}[\psi _{g}]=\int _{\mathbb {R} ^{3}}\psi _{g}^{*}\,{\hat {H}}\psi _{g}\,dV\geq E^{(0)}}$,

with equality holding if and only if ${\displaystyle \psi _{g}=\psi ^{(0)}}$. Thus, by minimizing ${\displaystyle E(c_{1},\ldots ,c_{n})={\mathcal {E}}[\psi _{g}]}$ with respect to coefficients ${\displaystyle c_{i}}$ for normalized trial wavefunctions ${\displaystyle \psi _{g}(c_{1},\ldots ,c_{n})}$, we obtain a closer approximation of the true ground-state wavefunction and its energy.

To start, we apply the normalization condition to the ansatz and expand to get an expression for N in terms of the ${\displaystyle c_{i}}$. Then, we substitute the ansatz into the expression for E and expand, yielding

${\displaystyle E(c_{1},\ldots ,c_{n})=N^{2}{\Big [}\sum _{i=1}^{n}c_{i}^{2}H_{ii}+\sum _{1\leq i\neq j\leq n}c_{i}c_{j}H_{ij}{\Big ]}}$, where ${\displaystyle N={\Big [}\sum _{i=1}^{n}c_{i}^{2}S_{ii}+\sum _{1\leq i\neq j\leq n}c_{i}c_{j}S_{ij}{\Big ]}^{-1/2}}$,

${\displaystyle S_{ij}=\int _{\mathbb {R} ^{3}}\phi _{i}^{*}\,\phi _{j}\,dV}$, and ${\displaystyle H_{ij}=\int _{\mathbb {R} ^{3}}\phi _{i}^{*}\,{\hat {H}}\phi _{j}\,dV}$.

In the remainder of the derivation, we will assume that the atomic orbitals are real. (For the simple case of the Hückel theory, they will be the 2p_z orbitals on carbon.) Thus, ${\displaystyle S_{ij}=S_{ji}^{*}=S_{ji}}$, and because the Hamiltonian operator is hermitian, ${\displaystyle H_{ij}=H_{ji}^{*}=H_{ji}}$. Setting ${\displaystyle \partial {E}/\partial {c_{i}}=0}$ for ${\displaystyle i=1,\ldots ,n}$ to minimize E and collecting terms, we obtain a system of n simultaneous equations

${\displaystyle \sum _{j=1}^{n}c_{j}(H_{ij}-ES_{ij})=0\quad (i=1,\cdots ,n)}$.

When ${\displaystyle i\neq j}$, ${\displaystyle S_{ij}}$ and ${\displaystyle H_{ij}}$ are called the overlap and resonance (or exchange) integrals, respectively, while ${\displaystyle H_{ii}}$ is called the Coulomb integral, and ${\displaystyle S_{ii}=1}$ simply expresses that fact that the ${\displaystyle \phi _{i}}$ are normalized. The n × n matrices ${\displaystyle [S_{ij}]}$ and ${\displaystyle [H_{ij}]}$ are known as the overlap and Hamiltonian matrices, respectively.

By a well-known result from linear algebra, finding nontrivial solutions to the simultaneous equations can be achieved by finding values of ${\displaystyle E}$ such that

${\displaystyle \mathrm {det} ([H_{ij}-ES_{ij}])=0}$. (*)

This determinant expression is known as the secular determinant. The variational theorem guarantees that the lowest value of ${\displaystyle E}$ that gives rise to a nontrivial (that is, not all zero) solution set ${\displaystyle (c_{1},c_{2},\ldots ,c_{n})}$ represents the best LCAO approximation of the energy of the most stable π orbital; higher values of ${\displaystyle E}$ with nontrivial solution sets represent reasonable estimates of the energies of the remaining π orbitals.

The Hückel method makes a few further simplifying assumptions concerning the values of the ${\displaystyle S_{ij}}$ and ${\displaystyle H_{ij}}$. In particular, it is first assumed that distinct ${\displaystyle \phi _{i}}$ have zero overlap. Together with the assumption that ${\displaystyle \phi _{i}}$ are normalized, this means that the overlap matrix is the n × n identity matrix: ${\displaystyle [S_{ij}]=\mathbf {I} _{n}}$. The generalized eigenvalue problem (*) then reduces to finding the eigenvalues of the Hamiltonian matrix ${\displaystyle [H_{ij}]}$.

Second, in the simplest case of a planar, unsaturated hydrocarbon, the Hamiltonian matrix ${\displaystyle \mathbf {H} =[H_{ij}]}$ is parameterized in the following way:

${\displaystyle H_{ij}={\begin{cases}\alpha ,&i=j;\\\beta ,&i,j\ \ {\text{adjacent}};\\0,&{\text{otherwise}}.\end{cases}}}$ (**)

Thus, we are assuming that: (1) the energy of an electron in an isolated C(2p_z) orbital is ${\displaystyle H_{ii}=\alpha }$; (2) the energy of interaction between C(2p_z) orbitals on adjacent carbons i and j (i.e., i and j are connected by a σ-bond) is ${\displaystyle H_{ij}=\beta }$; (3) orbitals on carbons not joined in this way are assumed not to interact, so ${\displaystyle H_{ij}=0}$ for nonadjacent i and j; and, as mentioned above, (4) the spatial overlap of electron density between different orbitals, represented by non-diagonal elements of the overlap matrix, is ignored by setting ${\displaystyle S_{ij}=0\ \ (i\neq j)}$, even when the orbitals are adjacent.

This neglect of orbital overlap is an especially severe approximation. For typical bond distances (1.40 Å) as might be found in benzene, for example, the true value of the overlap for C(2p_z) orbitals on adjacent atoms i and j is about ${\displaystyle S_{ij}=0.21}$; even larger values are found when the bond distance is shorter (e.g., ${\displaystyle S_{ij}=0.27}$ ethylene).^[12] A major consequence of having nonzero overlap integrals is the fact that, compared to non-interacting isolated orbitals, bonding orbitals are not energetically stabilized by nearly as much as antibonding orbitals are destabilized. The orbital energies derived from the Hückel treatment do not account for this asymmetry (see Hückel solution for ethylene (below) for details).

The eigenvalues of ${\displaystyle \mathbf {H} }$ are the Hückel molecular orbital energies ${\displaystyle E_{1},\ldots ,E_{n}}$, expressed in terms of ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, while the eigenvectors are the Hückel MOs ${\displaystyle \Psi _{1},\ldots ,\Psi _{n}}$, expressed as linear combinations of the atomic orbitals ${\displaystyle \phi _{i}}$. Using the expression for the normalization constant N and the fact that ${\displaystyle [S_{ij}]=\mathbf {I} _{n}}$, we can find the normalized MOs by incorporating the additional condition

${\displaystyle \sum _{i=1}^{n}c_{i}^{2}=1}$.

The Hückel MOs are thus uniquely determined when eigenvalues are all distinct. When an eigenvalue is degenerate (two or more of the ${\displaystyle E_{i}}$ are equal), the eigenspace corresponding to the degenerate energy level has dimensionality greater than 1, and the normalized MOs at that energy level are then not uniquely determined. When that happens, further assumptions pertaining to the coefficients of the degenerate orbitals (usually ones that make the MOs orthogonal and mathematically convenient^[13]) have to be made in order to generate a concrete set of molecular orbital functions.

If the substance is a planar, unsaturated hydrocarbon, the coefficients of the MOs can be found without appeal to empirical parameters, while orbital energies are given in terms of only ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. On the other hand, for systems containing heteroatoms, such as pyridine or formaldehyde, values of correction constants ${\displaystyle h_{\mathrm {X} }}$ and ${\displaystyle k_{\mathrm {X-Y} }}$ have to be specified for the atoms and bonds in question, and ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ in (**) are replaced by ${\displaystyle \alpha +h_{\mathrm {X} }\beta }$ and ${\displaystyle k_{\mathrm {X-Y} }\beta }$, respectively.

Hückel solution for ethylene

Molecular orbitals ethylene ${\displaystyle E=\alpha -\beta }$

In the Hückel treatment for ethylene, we write the Hückel MOs ${\displaystyle \Psi \,}$ as a linear combination of the atomic orbitals (2p orbitals) on each of the carbon atoms:

${\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}}$.

Molecular orbitals ethylene ${\displaystyle E=\alpha +\beta }$

Applying the result obtained by the Ritz method, we have the system of equations

${\displaystyle {\begin{bmatrix}H_{11}-ES_{11}&H_{12}-ES_{12}\\H_{21}-ES_{21}&H_{22}-ES_{22}\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$,

where:

${\displaystyle H_{ij}=\int \phi _{i}\,{\hat {H}}\phi _{j}\,dV}$ and

${\displaystyle S_{ij}=\int \phi _{i}\,\phi _{j}\,dV}$.

(Since 2p_z atomic orbital can be expressed as a pure real function, the * representing complex conjugation can be dropped.) The Hückel method assumes that all overlap integrals (including the normalization integrals) equal the Kronecker delta, ${\displaystyle S_{ij}=\delta _{ij}\,}$, all Coulomb integrals ${\displaystyle H_{ii}\,}$ are equal, and the resonance integral ${\displaystyle H_{ij}\,}$ is nonzero when the atoms i and j are bonded. Using the standard Hückel variable names, we set

${\displaystyle H_{11}=H_{22}=\alpha \,}$,

${\displaystyle H_{12}=H_{21}=\beta \,}$,

${\displaystyle S_{11}=S_{22}=1\,}$, and

${\displaystyle S_{12}=S_{21}=0\,}$.

The Hamiltonian matrix is

${\displaystyle \mathbf {H} ={\begin{bmatrix}\alpha &\beta \\\beta &\alpha \\\end{bmatrix}}}$.

The matrix equation that needs to be solved is then

${\displaystyle {\begin{bmatrix}\alpha -E&\beta \\\beta &\alpha -E\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$,

or, dividing by ${\displaystyle \beta }$,

${\displaystyle {\begin{bmatrix}{\frac {\alpha -E}{\beta }}&1\\1&{\frac {\alpha -E}{\beta }}\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$.

Setting ${\displaystyle x:={\frac {\alpha -E}{\beta }}}$, we obtain

${\displaystyle {\begin{bmatrix}x&1\\1&x\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\\end{bmatrix}}=0}$. (***)

This homogeneous system of equations have nontrivial solutions for ${\displaystyle c_{1},c_{2}}$ (solutions besides the physically meaningless ${\displaystyle c_{1}=c_{2}=0}$) iff the matrix is singular and the determinant is zero:

${\displaystyle {\begin{vmatrix}x&1\\1&x\\\end{vmatrix}}=0}$.

Solving for ${\displaystyle x}$,

${\displaystyle x^{2}-1=0\,}$, or

${\displaystyle x=\pm 1\,}$.

Since ${\displaystyle E=\alpha -x\beta }$, the energy levels are

${\displaystyle E=\alpha -\pm 1\times \beta }$, or

${\displaystyle E=\alpha \mp \beta }$.

The coefficients can then be found by expanding (***):

${\displaystyle c_{2}=-xc_{1}\,}$and

${\displaystyle c_{1}=-xc_{2}\,}$.

Since the matrix is singular, the two equations are linearly dependent, and the solution set is not uniquely determined until we apply the normalization condition. We can only solve for ${\displaystyle c_{2}}$ in terms of ${\displaystyle c_{1}}$:

${\displaystyle c_{2}=-\pm 1\times c_{1}\,}$, or

${\displaystyle c_{2}=\mp c_{1}\,}$.

After normalization with ${\displaystyle c_{1}^{2}+c_{2}^{2}=1}$, the numerical values of ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$ can be found:

${\displaystyle c_{1}={\frac {1}{\sqrt {2}}}}$ and ${\displaystyle c_{2}=\mp {\frac {1}{\sqrt {2}}}}$.

Finally, the Hückel molecular orbitals are

${\displaystyle \Psi _{\mp }=c_{1}\phi _{1}+c_{2}\phi _{2}={\frac {1}{\sqrt {2}}}\phi _{1}\mp {\frac {1}{\sqrt {2}}}\phi _{2}={\frac {\phi _{1}\mp \phi _{2}}{\sqrt {2}}}\,}$.

The constant β in the energy term is negative; therefore, ${\displaystyle E_{+}=\alpha +\beta }$ with ${\textstyle \Psi _{+}={\frac {1}{\sqrt {2}}}(\phi _{1}+\phi _{2})\,}$ is the lower energy corresponding to the HOMO energy and ${\displaystyle E_{-}=\alpha -\beta }$ with ${\textstyle \Psi _{-}={\frac {1}{\sqrt {2}}}(\phi _{1}-\phi _{2})\,}$ is the LUMO energy.

If, contrary to the Hückel treatment, a positive value for ${\displaystyle S:=S_{12}=S_{21}}$ were included, the energies would instead be

${\displaystyle E_{\pm }={\frac {\alpha \pm \beta }{1\pm S}}}$,

while the corresponding orbitals would take the form

${\displaystyle \Psi _{\pm }={\sqrt {\frac {1}{2\pm 2S}}}\phi _{1}\pm {\sqrt {\frac {1}{2\pm 2S}}}\phi _{2}}$.

An important consequence of setting ${\displaystyle S>0}$ is that the bonding (in-phase) combination is always stabilized to a lesser extent than the antibonding (out-of-phase) combination is destabilized, relative to the energy of the free 2p orbital. Thus, in general, 2-center 4-electron interactions, where both the bonding and antibonding orbitals are occupied, are destabilizing overall. This asymmetry is ignored by Hückel theory. In general, for the orbital energies derived from Hückel theory, the sum of stabilization energies for the bonding orbitals is equal to the sum of destabilization energies for the antibonding orbitals, as in the simplest case of ethylene shown here and the case of butadiene shown below.

Hückel solution for 1,3-butadiene

Butadiene molecular orbitals

Similarly, in the Hückel treatment for 1,3-butadiene, we write the molecular orbital ${\displaystyle \Psi \,}$ as a linear combination of the four atomic orbitals ${\displaystyle \phi _{i}}$ (carbon 2p orbitals) with coefficients ${\displaystyle c_{i}}$:

${\displaystyle \ \Psi =c_{1}\phi _{1}+c_{2}\phi _{2}+c_{3}\phi _{3}+c_{4}\phi _{4}}$.

The Hamiltonian matrix is

${\displaystyle \mathbf {H} ={\begin{bmatrix}\alpha &\beta &0&0\\\beta &\alpha &\beta &0\\0&\beta &\alpha &\beta \\0&0&\beta &\alpha \\\end{bmatrix}}}$.

In the same way, we write the secular equations in matrix form as

${\displaystyle {\begin{bmatrix}\alpha -E&\beta &0&0\\\beta &\alpha -E&\beta &0\\0&\beta &\alpha -E&\beta \\0&0&\beta &\alpha -E\\\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\\c_{4}\\\end{bmatrix}}=0}$,

which leads to

${\displaystyle {\Big (}{\frac {\alpha -E}{\beta }}{\Big )}^{4}-3{\Big (}{\frac {\alpha -E}{\beta }}{\Big )}^{2}+1=0}$

and

${\displaystyle E_{1,2,3,4}=\alpha +{\frac {{\sqrt {5}}\pm 1}{2}}\beta ,\alpha -{\frac {{\sqrt {5}}\mp 1}{2}}\beta }$, or approximately,

${\displaystyle E_{1,2,3,4}\approx \alpha +1.618\beta ,\alpha +0.618\beta ,\alpha -0.618\beta ,\alpha -1.618\beta }$, where 1.618... and 0.618... are the golden ratios ${\displaystyle \varphi }$ and ${\displaystyle 1/\varphi }$.

The orbitals are given by

${\displaystyle \Psi _{1}\approx 0.372\phi _{1}+0.602\phi _{2}+0.602\phi _{3}+0.372\phi _{4}}$,

${\displaystyle \Psi _{2}\approx 0.602\phi _{1}+0.372\phi _{2}-0.372\phi _{3}-0.602\phi _{4}}$,

${\displaystyle \Psi _{3}\approx 0.602\phi _{1}-0.372\phi _{2}-0.372\phi _{3}+0.602\phi _{4}}$, and

${\displaystyle \Psi _{4}\approx 0.372\phi _{1}-0.602\phi _{2}+0.602\phi _{3}-0.372\phi _{4}}$.

Crystal field theory

From Wikipedia, the free encyclopedia

Crystal Field Theory (CFT) is a model that describes the breaking of degeneracies of electron orbital states, usually d or f orbitals, due to a static electric field produced by a surrounding charge distribution (anion neighbors). This theory has been used to describe various spectroscopies of transition metal coordination complexes, in particular optical spectra (colors). CFT successfully accounts for some magnetic properties, colors, hydration enthalpies, and spinel structures of transition metal complexes, but it does not attempt to describe bonding. CFT was developed by physicists Hans Bethe and John Hasbrouck van Vleck^[1] in the 1930s. CFT was subsequently combined with molecular orbital theory to form the more realistic and complex ligand field theory (LFT), which delivers insight into the process of chemical bonding in transition metal complexes.

Overview of crystal field theory analysis

According to crystal field theory, the interaction between a transition metal and ligands arises from the attraction between the positively charged metal cation and negative charge on the non-bonding electrons of the ligand. The theory is developed by considering energy changes of the five degenerate d-orbitals upon being surrounded by an array of point charges consisting of the ligands. As a ligand approaches the metal ion, the electrons from the ligand will be closer to some of the d-orbitals and farther away from others, causing a loss of degeneracy. The electrons in the d-orbitals and those in the ligand repel each other due to repulsion between like charges. Thus the d-electrons closer to the ligands will have a higher energy than those further away which results in the d-orbitals splitting in energy. This splitting is affected by the following factors:

• the nature of the metal ion.
• the metal's oxidation state. A higher oxidation state leads to a larger splitting relative to the spherical field.
• the arrangement of the ligands around the metal ion.
• the nature of the ligands surrounding the metal ion. The stronger the effect of the ligands then the greater the difference between the high and low energy d groups.

The most common type of complex is octahedral; here six ligands form an octahedron around the metal ion. In octahedral symmetry the d-orbitals split into two sets with an energy difference, Δ_oct (the crystal-field splitting parameter) where the d_xy, d_xz and d_yz orbitals will be lower in energy than the d_z² and d_x²-y², which will have higher energy, because the former group is farther from the ligands than the latter and therefore experience less repulsion. The three lower-energy orbitals are collectively referred to as t_2g, and the two higher-energy orbitals as e_g. (These labels are based on the theory of molecular symmetry). Typical orbital energy diagrams are given below in the section High-spin and low-spin.

Tetrahedral complexes are the second most common type; here four ligands form a tetrahedron around the metal ion. In a tetrahedral crystal field splitting, the d-orbitals again split into two groups, with an energy difference of Δ_tet. The lower energy orbitals will be d_z² and d_x²-y², and the higher energy orbitals will be d_xy, d_xz and d_yz - opposite to the octahedral case. Furthermore, since the ligand electrons in tetrahedral symmetry are not oriented directly towards the d-orbitals, the energy splitting will be lower than in the octahedral case. Square planar and other complex geometries can also be described by CFT.

The size of the gap Δ between the two or more sets of orbitals depends on several factors, including the ligands and geometry of the complex. Some ligands always produce a small value of Δ, while others always give a large splitting. The reasons behind this can be explained by ligand field theory. The spectrochemical series is an empirically-derived list of ligands ordered by the size of the splitting Δ that they produce (small Δ to large Δ; see also this table):

I⁻ < Br⁻ < S²⁻ < SCN⁻ (S–bonded) < Cl⁻ < NO₃⁻ < N₃⁻ < F⁻ < OH⁻ < C₂O₄²⁻ < H₂O < NCS⁻ (N–bonded) < CH₃CN < py < NH₃ < en < 2,2'-bipyridine < phen < NO₂⁻ < PPh₃ < CO < CN⁻.

It is useful to note that the ligands producing the most splitting are those that can engage in metal to ligand back-bonding.

The oxidation state of the metal also contributes to the size of Δ between the high and low energy levels. As the oxidation state increases for a given metal, the magnitude of Δ increases. A V³⁺ complex will have a larger Δ than a V²⁺ complex for a given set of ligands, as the difference in charge density allows the ligands to be closer to a V³⁺ ion than to a V²⁺ ion. The smaller distance between the ligand and the metal ion results in a larger Δ, because the ligand and metal electrons are closer together and therefore repel more.

High-spin and low-spin

Low Spin [Fe(NO₂)₆]³⁻ crystal field diagram

Ligands which cause a large splitting Δ of the d-orbitals are referred to as strong-field ligands, such as CN⁻ and CO from the spectrochemical series. In complexes with these ligands, it is unfavourable to put electrons into the high energy orbitals. Therefore, the lower energy orbitals are completely filled before population of the upper sets starts according to the Aufbau principle. Complexes such as this are called "low spin". For example, NO₂⁻ is a strong-field ligand and produces a large Δ. The octahedral ion [Fe(NO₂)₆]³⁻, which has 5 d-electrons, would have the octahedral splitting diagram shown at right with all five electrons in the t_2g level. The low spin state therefore does not follow Hund's rule.

High Spin [FeBr₆]³⁻ crystal field diagram

Conversely, ligands (like I⁻ and Br⁻) which cause a small splitting Δ of the d-orbitals are referred to as weak-field ligands. In this case, it is easier to put electrons into the higher energy set of orbitals than it is to put two into the same low-energy orbital, because two electrons in the same orbital repel each other. So, one electron is put into each of the five d-orbitals before any pairing occurs in accord with Hund's rule and "high spin" complexes are formed. For example, Br⁻ is a weak-field ligand and produces a small Δ_oct. So, the ion [FeBr₆]³⁻, again with five d-electrons, would have an octahedral splitting diagram where all five orbitals are singly occupied.

In order for low spin splitting to occur, the energy cost of placing an electron into an already singly occupied orbital must be less than the cost of placing the additional electron into an e_g orbital at an energy cost of Δ. As noted above, e_g refers to the d_z² and d_x²-y² which are higher in energy than the t_2g in octahedral complexes. If the energy required to pair two electrons is greater than Δ, the energy cost of placing an electron in an e_g, high spin splitting occurs.

The crystal field splitting energy for tetrahedral metal complexes (four ligands) is referred to as Δ_tet, and is roughly equal to 4/9Δ_oct (for the same metal and same ligands). Therefore, the energy required to pair two electrons is typically higher than the energy required for placing electrons in the higher energy orbitals. Thus, tetrahedral complexes are usually high-spin.

The use of these splitting diagrams can aid in the prediction of magnetic properties of coordination compounds. A compound that has unpaired electrons in its splitting diagram will be paramagnetic and will be attracted by magnetic fields, while a compound that lacks unpaired electrons in its splitting diagram will be diamagnetic and will be weakly repelled by a magnetic field.

Crystal field stabilization energy

The crystal field stabilization energy (CFSE) is the stability that results from placing a transition metal ion in the crystal field generated by a set of ligands. It arises due to the fact that when the d-orbitals are split in a ligand field (as described above), some of them become lower in energy than before with respect to a spherical field known as the barycenter in which all five d-orbitals are degenerate. For example, in an octahedral case, the t_2g set becomes lower in energy than the orbitals in the barycenter. As a result of this, if there are any electrons occupying these orbitals, the metal ion is more stable in the ligand field relative to the barycenter by an amount known as the CFSE. Conversely, the e_g orbitals (in the octahedral case) are higher in energy than in the barycenter, so putting electrons in these reduces the amount of CFSE.

Octahedral crystal field stabilization energy

If the splitting of the d-orbitals in an octahedral field is Δ_oct, the three t_2g orbitals are stabilized relative to the barycenter by ²/₅ Δ_oct, and the e_g orbitals are destabilized by ³/₅ Δ_oct. As examples, consider the two d⁵ configurations shown further up the page. The low-spin (top) example has five electrons in the t_2g orbitals, so the total CFSE is 5 x ²/₅ Δ_oct = 2Δ_oct. In the high-spin (lower) example, the CFSE is (3 x ²/₅ Δ_oct) - (2 x ³/₅ Δ_oct) = 0 - in this case, the stabilization generated by the electrons in the lower orbitals is canceled out by the destabilizing effect of the electrons in the upper orbitals.

Optical properties

The optical properties (details of absorption and emission spectra) of many coordination complexes can be explained by Crystal Field Theory. Often, however, the deeper colors of metal complexes arise from more intense charge-transfer excitations.^[2]

Geometries and crystal field splitting diagrams

Name	Shape	Energy diagram
Octahedral
Pentagonal bipyramidal
Square antiprismatic
Square planar
Square pyramidal
Tetrahedral
Trigonal bipyramidal

The Marshmallow Myth

New research suggests that delayed gratification is overrated.

Posted Mar 09, 2017

Author:  Nick Tasler

Original post:  https://www.psychologytoday.com/intl/blog/strategic-thinking/201703/the-marshmallow-myth

It's been over half a century since Walter Mischel tempted the tastebuds of hundreds of children in Northern California with a tantalizing marshmallow treat inside his laboratory at Stanford University. In the decades that followed, the children who were able to resist eating a smaller treat right away in exchange for a bigger treat later on ended up being healthier and more successful in school and in life.

The message was clear: delaying gratification is a critical success factor. Thus prompting teachers and parents all across America to sport "Don't Eat the Marshmallow" tee-shirts.

But new evidence suggests that we are missing the point.

DARK SIDE OF THE MARSHMALLOW

In a series of five studies recently published in Personality and Social Psychology Bulletin (link is external), researchers Kaitlin Wooley and Ayelet Fishbach at the University of Chicago's Booth School of Business found that the experience of immediate rewards—such as enjoying the taste of a healthy food—predicted, more strongly than anticipated rewards did, how persistent people would be in pursuit of their goals to exercise more; study longer; eat healthier; stick with a new year's resolution; or sustain a lifestyle change.

It turns out that long-term desires like making the honor roll, getting a promotion, or fitting into a smaller pair of pants fuel the motivation to set goals in the first place. But after we define that future vision of where we wish to end up, reminding ourselves how badly we want to get there—how much we really want to squeeze into those skinny jeans or earn that raise—does a relatively poor job of keeping us motivated to resist temptation for the weeks or months it will take to achieve the goal.

The Chicago studies found that those who succeed at doing something new are not just those who are better at delaying gratification. Those who succeed are better finding other ways to gratify themselves until they reach that bigger goal.

Instead of simply grinning and bearing the misery of jogging, people who successfully meet their goal of exercising more are the ones who switch to Zumba or find a jogging partner they like talking to everyday.

The grittiest college students aren't those who constantly sacrifice pleasure by imagining the day they'll finally get to become an investment banker. They are the students who focus on the satisfaction they feel every time they accumulate a new piece of knowledge or on the immediate pride they feel each time they crack open a book instead of a beer.

This also explains Teresa Amabile's and Steven Kramer's  (link is external)discovery that the number one predictor of work engagement is a phenomenon they call "the progress principle." At work, we throw ourselves into challenging projects not because our boss blankets us with warm fuzzies or because we think it will add another zero to the east side of our paycheck. More than anything else, people stay engaged in hard work when they feel like they are making progress on a project that matters.

The Chicago studies tell us why. By setting and achieving tiny goals every couple of days, we tap into a constant flow of immediate gratification needed to keep us motivated in pursuit of that distant goal.

In the science of gratification, the cliché holds true: It really is more about enjoying the journey rather than imagining the destination.

Then again, isn't this truly what Walter Mischel found?

REDEEMING THE MARSHMALLOW

The kids in Mischel's studies at Stanford and later in the South Bronx who successfully "passed" the marshmallow test, were the kids who distracted themselves from the mouth-watering sugar puff in front of them.

And how did they distract themselves? They sang a song or played a game inside their heads. What they did not do was sit there and stare at the marshmallow, vainly attempting to tap into deep reservoirs of self-control or personal discipline.

So we have to ask ourselves: Is "delaying gratification" what the successful kids actually did? Or did they merely substitute the source of their gratification—switching their focus from something possibly gratifying in the future (more marshmallows) to something immediately gratifying in the present (singing and playing)?

To say that successful kids and adults "delay gratification" misses the point.

When we focus on "not eating the marshmallow" we imply that successful people are those who are better at sucking it up and enduring the pain and struggle necessary to pursue an expected gain some day, week, or year down the road. Meanwhile, they let all the pleasures of life pass them by.

When we set up the situation this way for our kids, our employees, or ourselves, we are presenting life as a choice between either a) succeed later, but be miserable now, or b) enjoy today even though it means sacrificing your future.

Could this be why so many of us toggle back and forth between feeling guilty one minute about our inability to stick with our goals, and then the very next minute feeling just as guilty about our inability to "be present" and to "live in the now"? No matter which choice you make, you lose.

But that's the exciting thing about the Chicago studies. They show us that this is a false choice.

The secret to success in the marshmallow test of life and work is not about delaying gratification. It is about discovering gratification in every situation. It's about leveraging the unparalleled ability of the human mind to find—and focus on—small sources of gratification in any set of circumstances. 

The UN Admits That The Paris Climate Deal Was A Fraud

• 11/02/2017
• Original post:  https://www.investors.com/politics/editorials/the-un-admits-that-the-paris-climate-deal-was-a-fraud/

Global Hot Air: Here's a United Nations climate report that environmentalists probably don't want anybody to read. It says that even if every country abides by the grand promises they made last year in Paris to reduce greenhouse gases, the planet would still be "doomed."

When President Obama hitched America to the Paris accords in 2016, he declared that it was "the moment that we finally decided to save our planet." And when Trump pulled out of the deal this year, he was berated by legions of environmentalists for killing it.

But it turns out that the Paris accord was little more than a sham that will do nothing to "save the planet."

According to the latest annual UN report on the "emissions gap," the Paris agreement will provide only a third of the cuts in greenhouse gas that environmentalists claim is needed to prevent catastrophic warming. If every country involved in those accords abides by their pledges between now and 2030 — which is a dubious proposition — temperatures will still rise by 3 degrees C by 2100. The goal of the Paris agreement was to keep the global temperature increase to under 2 degrees.

Eric Solheim, head of the U.N. Environment Program, which produces the annual report, said this week that "One year after the Paris Agreement entered into force, we still find ourselves in a situation where we are not doing nearly enough to save hundreds of millions of people from a miserable future. Governments, the private sector and civil society must bridge this catastrophic climate gap."

The report says unless global greenhouse gas emissions peak before 2020, the CO2 levels will be way above the goal set for 2030, which, it goes on, will make it "extremely unlikely that the goal of holding global warming to well below 2 degrees C can still be reached."

Not to worry. The UN claims that closing this gap will be easy enough, if nations set their collective minds to it.

But this is a fantasy. The list of what would need to be done by 2020 — a little over two years from now — includes: Boosting renewable energy's share to 30%. Pushing electric cars to 15% of new car sales, up from less than 1% today. Doubling mass transit use. Cutting air travel CO2 emissions by 20%. And coming up with $1 trillion for "climate action."

Oh, and coal-fired power plants would have to be phased out worldwide, starting now.

According to the report, "phasing out coal consumption … is an indispensable condition for achieving international climate change targets." That means putting a halt to any new coal plants while starting to phase out the ones currently in use.

Good luck with that. There are currently 273 gigawatts of coal capacity under construction around the world, and another 570 gigawatts in the pipeline, the UN says. That would represent a 42% increase in global energy production from coal. Does anyone really think developing countries who need coal as a cheap source of fuel to grow their economies will suddenly call it quits?

So, does this mean the planet is doomed? Hardly. As we have noted in this space many times, all those forecasts of global catastrophe are based on computer models that have been unreliable predictors of warming. And all of the horror stories assume the worst.

What the report does make clear, however, is that all the posturing by government leaders in Paris was just that. Posturing. None of these countries intended to take the drastic and economically catastrophic steps environmentalist claim are needed to prevent a climate change doomsday.  As such, Trump was right to stop pretending.

Whether you believe in climate change or not, the Paris climate accord amounted to nothing, or pretty close to it. Even the UN admits that now.

Lakes of liquid CO2 in the deep sea

Kenneth Nealson^*

Original post:  https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1599885/

The thought of liquid CO₂ conjures up different things to different folks: perhaps the decaffeination of coffee beans, perhaps the recently popularized “green” method for dry cleaning, or even phase diagrams that occupied a part of one's life in past chemistry classes. What it does not conjure up is a subsurface lake at the bottom of the ocean, a lake with abundant living microbes, as reported in this issue of PNAS by Inagaki et al. (1). These authors discovered such a place near the Yonaguni Knoll in the Okinawa Trough at a depth of ≈1,400 m. The description in both words and video (see supporting movie 1 in ref. 1) is quite striking. First, because liquid CO₂ at this depth is less dense than water (2, 3), so that such a lake should not be present. Second, because this is a phenomenon that few of us have ever seen, movie 1 in ref. 1 reveals a flowing stream of liquid CO₂ that seems almost surreal.

The answer to the apparent conundrum surrounding the very existence of this phenomenon is that the lake is maintained in place by a surface pavement and a subpavement cap of CO₂ hydrate (CO₂·6H₂O) that traps the low-density liquid CO₂ in place. At the temperature of the seafloor at this depth, such a CO₂ hydrate should be stable (4), leading to a structure similar to that shown in figure 1 of the Inagaki et al. article (1), in which a surface pavement overlies a layer of CO₂ hydrate that serves as a cap for the subsurface lake. The surface pavement is quite remarkable, having a very unusual elemental sulfur content of >50%. It may well be that there are clues to the origin of the sulfur (and the role of sulfur metabolism in this system) in both the isotopic composition of the sulfur and the chemical and biological nature of the “sulfur-hydrate complex,” things that should be resolved in future studies. As discussed in a another recent article in PNAS (3), the density of liquid CO₂ increases with depth, so that at depths of 3,000–3,800 m (density reaches a maximum at ≈3,500 m and decreases at greater depth), it is more dense than seawater [see figure 2 in House et al. (3)], forming a natural negative buoyancy zone, where one could rightfully expect to see lakes of liquid CO₂. CO₂ hydrates also form at these depths, suggesting that large subsurface lakes of liquid CO₂ capped by hydrates could be excellent locations for the large-scale injection and disposal of CO₂ (3). The notion that similar sites might exist as natural systems was not entertained in the House et al. article, but if they do, and are stable in the long term, then the notion that communities of microbes might be capable of adapting to such an environment becomes of great interest. Such knowledge also becomes of importance with regard to the establishment of such reservoirs in the deep sea.

Liquid CO₂ in the deep ocean is not an unprecedented finding. In 1990, Sakai et al. (4) noted the release of CO₂ droplets at a depth of 1,400 m and a temperature of 3.8°C in a region near the mid-Okinawa Trough, and more recently, similar observations were made in the northern Mariana Arc (5). What is new is the concept that large bodies of liquid CO₂ may exist as subsurface lakes in such zones. For example, the northern Mariana is a volcanic arc with little or no sediment deposition. Thus, one does not expect to find sediment-hosted lakes such as are reported by Inagaki et al. (1). How many such “lakes” are there? How stable are they, and are they potential players in the global carbon cycle? Given our paucity of knowledge about such systems, it is fair to say that these questions remain unanswered. It may be of great interest to answer such questions for a variety of different reasons, as outlined below.

First, one of the proposed methods for disposal of CO₂ (and amelioration of the associated effects on global climate) is the direct injection of CO₂ into the deep sea (6–8). The expense of moving large amounts of CO₂ to 3,000 m and deeper and the problems with rapidly injecting it at these depths could be substantial. Neither the biological (toxicity) nor the physical (effect on porewaters from injection of massive amounts of liquid CO₂) impacts are known (3).

Do the findings of Inagaki et al. (1) offer another potential avenue for CO₂ storage? Probably not. The robustness of these systems clearly depends on the formation and long-term stability of the CO₂ hydrate cap, something that may not be routine to achieve. Whereas in the deeper ocean, the hydrate cap should be stable and the underlying liquid CO₂ can migrate downward until it becomes neutrally buoyant and will then move only by diffusion (3), burial in zones where liquid CO₂ is less buoyant than water would almost certainly have to be deeper into the sediments themselves, thus becoming subject to temperature changes caused by the geothermal gradient. Thus, the CO₂ lakes reported (1) probably will not lead the way to a new avenue for shallow-water CO₂ disposal. Such environments do, however, offer accessible sites where some of the other impacts of the liquid CO₂/seawater interface on the environment (including the resident biota) can be studied as a naturally existing phenomenon.

The second issue at hand relates to the potential toxicity of CO₂ to various forms of marine life. Indeed, it has been argued from many points of view that injection of CO₂ may have dire consequences on the deep sea biota. It is argued that the combined effects of CO₂ itself and the lowered pH that goes along with it will have minor to major impacts on marine life, including microbial life (9). The potential impact of CO₂ in liquid form may be much less extreme (3), but for the moment this remains unknown. Given that liquid CO₂ is decidedly nonpolar and behaves like an organic solvent, it might be a rather harsh environment itself, regardless of any pH-associated effects. To this end, it should also be noted that both pH and alkalinity were measured onboard after degassing had occurred during recovery of the samples. Thus, one expects the in situ pH to be lower and the alkalinity to be higher, stressing the importance of in situ measurements in future studies.

Inagaki et al. (1) studied both the chemistry of the environment and the populations of microbes present in both the sediments above the liquid CO₂ environment and the CO₂ lake itself (Fig. 1). The pH of the environment changes very little as one proceeds from the overlying sediment to the CO₂–hydrate/liquid CO₂ interface, being stable at ≈6.5, whereas the alkalinity increases from 20 to 30 mmol/kg. Alongside these rather stable values, the authors observed that the cell number (as judged from microscopic counts) declined from ≈10⁹ per ml in the overlying pavement to 10⁷ per ml in the liquid CO₂ zone and then increased again in the zone below the liquid CO₂ (Fig. 1). Although a 100-fold decrease is a large drop, the fact that 10⁷ cells per ml of intact cells remain is quite remarkable, given the potentially hostile nature of this nonpolar solvent. What are these cells? What are they doing? Are these new and unusual organisms, or are they just survivors that can tolerate this environment? For the moment, these questions cannot all be answered. Molecular methods were used to identify the major microbial groups, which individually looked familiar in terms of their phylogentic affiliations; similar microbes have been seen in deep-sea and methane-seep environments (10–13). However, the combinations of microbes present (i.e., the community composition) did not appear to make a story that could be easily related to any other deep-sea site examined so far. That is, this report revealed the presence of archaea previously identified in zones of anaerobic methane oxidation (the so-called ANME-2c group), and the Eel-2 group of Deltaproteobacteria, associated with sulfate reduction but not previously known as major components of anaerobic methane-oxidizing consortia. Furthermore, no members of the DSS group, a group of sulfate-reducing Deltaproteobacteria that are usually found with ANME-2 cells (10, 11), were detected. Thus, if there is a consortium driving anaerobic methane oxidation, it may well be something new. With the members of this environment now partially identified, it should be possible in future work to examine the consortium by fluorescence in situ hybridization (FISH) and other similar techniques to resolve these issues.

Fig. 1.

Data from the CO₂ lake zone, showing the vertical dimensions of the lake and overlying sediment, its general properties in terms of temperature, pH, sulfate, chlorinity, and cell number, as determined by acridine orange direct counts (AODC). The pavement is located down slope from a large black smoker, at a water depth of ≈1,400 m (1).

At depths of 3,000–3,800 m, CO₂ is more dense than seawater.

To further complicate the story, although both methane consumption activity measurements and stable isotope analyses of bacterial and archael lipids suggested that methane oxidation and perhaps CO₂ fixation were occurring in the liquid CO₂ zone, neither the predicted organisms nor the genes coding for the needed enzymes (as detected by gene probes) were abundant in this environment. This, of course, raises the very exciting possibility that both the methane oxidation measured and the lipid fractionation observed are the result of activities of entirely new types of organisms that may have eluded detection by standard molecular probe analyses. The resolution of this possibility will await the application of more detailed studies, including fluorescence labeling, substrate labeling, stable isotopic probing, and metagenomic community analysis.

But where are the organisms residing in this fascinating environment? Given the nature of liquid CO₂, it would seem likely that the resident population is in fact taking advantage of the presence of water/hydrate interface as the niche of choice. Such a microhabitat should provide a microbe with an acceptable place for life. Although laboratory experiments could establish this as a possibility, it is clear that in situ studies will be needed to locate microbes in their natural habitats.

Inagaki et al. (1) end their article with an interesting speculation as to the potential for such environments to exist on other solar-system bodies. If follow-up in situ studies to this work show that microbial life is capable of existing in the liquid CO₂ domain and what kinds of metabolism are consistent with such a habitat, it could well open up a new area with regard to the search for life both on Earth and elsewhere. As the authors explain: “the Yonaguni Knoll is an exceptional natural laboratory for the study of consequences of CO₂ disposal as well as of natural CO₂ reservoirs as potential microbial habitats on early Earth and other celestial bodies” (1). This is one of those rare times when the statement, “this system deserves more study” is surely true, with the immediate issues at hand being the identification of the microbes in the various substrata and the determination of the activities actually occurring in the liquid CO₂ environment.