Opposite-spin electron correlation

In an electronic system it is essential to distinguish between the correlation of electrons with identical spin, Fermi correlation, from that of electrons with opposite spins. Coulomb correlation. The correlation term/(ri,r2) as defined in eqn (E7.10) will measure both types of correlation (McWeeny 1960). However, the limiting value of the correlation hole as expressed in eqn (E7.11) arises only from the correlation between electrons of the same spin, the Fermi hole. That part of/(r, r2) that refers to the correlation between electrons of opposite spin contributes zero when integrated over all space. If the co-... 

Proynov, Salahub, and coworkers have developed several correlation functionals [205 -209] starting from a GAUSSIAN model of the spherically-averaged pair distribution function for opposite-spin electrons... 

It could be suspected that for hydrocarbons, the influence of correlation is not as much important as for other compounds. Then, possibly, the ELI-q for the singlet-coupled electrons would be of minor importance for the bonding analysis. To examine the extent of opposite-spin electron pairing in case of the C-C and C-H bonds, the C3H6 was analyzed. The CISD calculation of the cyclopropane molecule was performed with the triple-zeta basis CCT using 18 electrons excited into 75 orbitals. The correlated 2-matrix was used to compute the electron density as well as the ELI (and the spin-pair composition). 

As we ve noted several times, Hartree-Fock theory provides an inadequate treatment of the correlation between the motions of the electrons within a molecular system, especially that arising between electrons of opposite spin. 

When Hartree-Fock theory fulfills the requirement that 4 be invarient with respect to the exchange of any two electrons by antisymmetrizing the wavefunction, it automatically includes the major correlation effects arising from pairs of electrons with the same spin. This correlation is termed exchange correlation. The motion of electrons of opposite spin remains uncorrelated under Hartree-Fock theory, however. 

In the Hartree-Fock model, where we take account of antisymmetry, it turns out that there is no correlation between the positions of electrons of opposite spin, yet,... 

Naively it may be expected that the correlation between pairs of electrons belonging to the same spatial MO would be the major part of the electron correlation. However, as the size of the molecule increases, the number of electron pairs belonging to different spatial MOs grows faster than those belonging to the same MO. Consider for example the valence orbitals for CH4. There are four intraorbital electron pairs of opposite spin, but there are 12 interorbital pairs of opposite spin, and 12 interorbital pairs of the same spin. A typical value for the intraorbital pair correlation of a single bond is 20kcal/ mol, while that of an interorbital pair (where the two MO are spatially close, as in CH4) is 1 kcal/mol. The interpair correlation is therefore often comparable to the intrapair contribution. 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

As mentioned in the start of Chapter 4, the correlation between electrons of parallel spin is different from that between electrons of opposite spin. The exchange energy is by definition given as a sum of contributions from the a and /3 spin densities, as exchange energy only involves electrons of the same spin. The kinetic energy, the nuclear-electron attraction and Coulomb terms are trivially separable. 

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

The correlation error can, of course, be defined with reference to the Hartree scheme but, in modem literature on electronic systems, one usually starts out from the Hartree-Fock approximation. This means that the main error is due to the neglect of the Coulomb correlation between electrons with opposite spins and, unfor-tunetely, we can expect this correlation error to be fairly large, since we force pairs of electrons with antiparallel spins together in the same orbital in space. The background for this pairing of the electrons is partly the classical formulation of the Pauli principle, partly the mathematical fact that a single determinant in such a case can... 

Our little table of accuracy shows that, in the Hartree scheme, there is a certain balance between the correlation errors for the two spin types which, in some cases, may lead to a cancellation of these errors with good theoretical results as a consequence. After the introduction of exchange, this balance is gone and, if the correlation between electrons with opposite spins is not taken into proper account, the final results may be influenced by this large... 

Since rv and rx here refer to electrons having opposite spins, the form III. 130 introduces correlation only for electrons with antiparallel spins. It is clear that, by using different functions fv /2.. . ., fn, one may get a still lower energy, but additional complications will occur, since the wave function III. 7 corresponds then to a mixture between different spin types (cf. Brickstock and Pople 1953). 

The general idea of using different orbitals for different spins" seems thus to render an important extension of the entire framework of the independent-particle model. There seem to be essential physical reasons for a comparatively large orbital splitting depending on correlation, since electrons with opposite spins try to avoid each other because of their mutual Coulomb repulsion, and, in systems with unbalanced spins, there may further exist an extra exchange polarization of the type emphasized by Slater. 

Kolos, W., J. Chem. Phys. 27, 591, Excitation energies of C2H4. The correlation between electrons with opposite spins is estimated by multiplying the usual orbital wave functions by the inter-electronic distance. 

The single-Slater determinant includes correlation between the motion of two electrons with parallel spins that avoid each other because of the exclusion principle (Szabo and Ostlund 1989), but correlation between the motion of electrons with opposite spin is neglected. The wave function of Eq. (3.2) does not prevent the two electrons from being at the same point in space, which is physically impossible. The Slater determinant wave function is therefore described as uncorrelated. 

Fe—S dimers, 38 443-445 map, four-iron clusters, 38 458 -functional theory, 38 423-467 a and b densities, 38 440 broken symmetry method, 38 425 conservation equation, 38 437 correlation for opposite spins and Coulomb hole, 38 439-440 electron densities, 38 436 exchange energy and Fermi hole, 38 438-439... 

To put it crudely, this correlation ensures that electrons of the same spin cannot be in the same place at the same time. Therefore this type of correlation makes the Coulombic repulsion energy between electrons of the same spin smaller than that between electrons of opposite spin. This is the reason why Hand s rule states that, an electronic state in which two electrons occupy different orbitals with the same spin is lower in energy than an electronic state in which the electrons occupy the orbitals, but with opposite spins. 

The major difficulty in wave function based calculations is that, starting from an independent-particle model, correlation between electrons of opposite spin must somehow be introduced into T. Inclusion of this type of electron correlation is essential if energies are to be computed with any degree of accuracy. How, through the use of multiconfigurational wave functions, correlation between electrons of opposite spin is incorporated into is the subject of Section 3.2.3. 

Inclusion of Electron Correlation. HF calculations, performed with basis sets so large that the calculations approach the HF limit for a particular molecule, still calculate total energies rather poorly. The reason is that, as already discussed, HF wave functions include no correlation between electrons of opposite spins. In order to include this type of correlation, multiconfigurational (MC) wave functions, like that in Eq. 5, must be used. 

Functionals. The difference between the Fock operator, in wave function based calculations and the analogous operator in DFT calculations is that the Coulomb and exchange operators in T are replaced in DFT by a functional of the electron density. In principle, this functional should provide an exact formula for computing the Coulombic interactions between an electron in a KS orbital and all the other electrons in a molecule. To be exact, this functional must include corrections to the Coulombic repulsion energy, computed directly from the electron density, for exchange between electrons of the same spin and correlation between electrons of opposite spin. 

Spin density is found in the molecular plane because of spin polarization, which is an effect arising from exchange correlation. The Fermi hole that surrounds the unpaired electron allows other electrons of the same spin to localize above and below the molecular plane slightly more than can electrons of opposite spin. Thus, if the unpaired electron is a, we would expect there to be a slight excess of density in the molecular plane as a result, the hyperfine splitting should be negative (see Section 9.1.3), and this is indeed the situation observed experimentally. An ROHF wave function, because it requires the spatial distribution of both spins in the doubly occupied orbitals to be identical, cannot represent this physically realistic situation. 

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