Electron Correlation Methods

The Hartree-Fock method generates solutions to the Schrodinger equation where the real electron-electron interaction is replaced by an average interaction (Chapter 3). In a sufficiently large basis set, the HF wave function is able to account for ~99% of the total energy, but the remaining ~1% is often very important for describing chemical phenomena. The difference in energy between the HF and the lowest possible energy in the given basis set is called the Electron Correlation (EC) energy.1 Physically, it corresponds to the motion of the electrons being correlated, i.e. on the average they are further apart than described by the HF wave function. As shown below, an unrestricted Hartree-Fock (UHF) type of wave function is, to a certain extent, able to include electron correlation. The proper reference for discussing electron correlation is therefore a restricted (RHF) or restricted open-shell (ROHF) wave function, although many authors use a UHF wave function for open-shell species. In the RHF case, all the electrons are paired in molecular orbitals. The two electrons in an MO occupy the same physical space, and differ only in the spin function. The spatial overlap between the orbitals of two such "pair"-electrons is (exactly) one, while the overlap between two electrons belonging to different pairs is (exactly) zero, owing to the ortho-normality of the MOs. The latter is not the same as saying that there is no repulsion between electrons in different MOs, since the electron-electron repulsion integrals involve products of MOs ((ff) = 0 for i ^ j, but (fifjlglfifj) and (ffglff) are not necessarily zero).

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 spins, but there are twelve interorbital pairs of opposite spins, and twelve interorbital pairs of same spin. A typical value for the intraorbital pair correlation of a single bond is ~80kJ/mol, while that of an interorbital pair (where the two

Introduction to Computational Chemistry, Second Edition. Frank Jensen. © 2007 John Wiley & Sons, Ltd

MOs are spatially close, as in CH4) is ~8kJ/mol. The interpair correlation is therefore often comparable to the intrapair contribution.

Since the correlation between opposite spins has both intra- and interorbital 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 correlation 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, there is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, and the corresponding phenomenon for electrons of same spin is the Fermi hole. This hole picture is discussed in more detail in connection with density functional theory in Chapter 6.

Another distinction is between dynamic and static electron correlation. The dynamic contribution is associated with the "instant" correlation between electrons, such as between those occupying the same spatial orbital. The static part is associated with electrons avoiding each other on a more "permanent" basis, such as those occupying different spatial orbitals. The latter is also sometimes called a near-degeneracy effect, as it becomes important for systems where different orbitals (configurations) have similar energies. The electron correlation in a helium atom is almost purely dynamic, while the correlation in the H2 molecule at the dissociation limit is purely static (here the bonding and antibonding MOs become degenerate). At the equilibrium distance for H2 the correlation is mainly dynamic (resembles the He atom), but this gradually changes to static correlation as the bond distance is increased. Similarly, the Be atom contains both static (near degeneracy of the 1s22s2 and 1s22p2 configurations) and dynamical correlation. There is therefore no clear-cut way of separating the two types of correlation, although they form a conceptually useful way of thinking about correlation effects.

The HF method determines the energetically best one-determinant trial wave function (within the given basis set). It is therefore clear that, in order to improve on HF results, the starting point must be a trial wave function that contains more than one Slater determinant (SD) O. This also means that the mental picture of electrons residing in orbitals has to be abandoned and the more fundamental property, the electron density, should be considered. As the HF solution usually gives ~99% of the correct answer, electron correlation methods normally use the HF wave function as a starting point for improvements.

A generic multi-determinant trial wave function can be written as in eq. (4.1), where a0 is usually close to one.

Electron correlation methods differ in how they calculate the coefficients in front of the other determinants, with a0 being determined by the normalization condition.

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO "function" in the "coordinate system" of the basis functions.The multi-determinant wave function (eq. (4.1)) can similarly be considered as describing the total wave function in a "coordinate" system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation).

CaXa a=l

Figure 4.1 Progression from atomic orbitals (AO) (basis functions), to molecular orbitals (MO), to Slater determinants (SD) and to a many-electron (ME) wave function

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