HF theory

One of the limitations of HF calculations is that they do not include electron correlation. This means that HF takes into account the average affect of electron repulsion, but not the explicit electron-electron interaction. Within HF theory the probability of finding an electron at some location around an atom is determined by the distance from the nucleus but not the distance to the other electrons as shown in Figure 3.1. This is not physically true, but it is the consequence of the central field approximation, which defines the HF method. 

FIGURE 3.1 Two arrangements of electrons around the nucleus of an atom having the same probability within HF theory, but not in correlated calculations. 

The basic physical idea of HF theory is a simple one and can be tied in very nicely with our discussion of the electron density given in Chapter 5. We noted the physical significance of the density function pi(r, 5) p (r, s)drdv gives the chance of finding any electron simultaneously in the spin-space volume elements dr and dr, with the other electrons anywhere in space and with either spin, / (r) dr gives the corresponding chance of finding any electron with either spin in the spatial volume element dr. 

I am assuming that this particular electronic state is the lowest-energy one of that given spatial symmetry, and that the i/f s are orthonormal. The first assumption is a vital one, the second just makes the algebra a little easier. The aim of HF theory is to find the best form of the one-electron functions i/ a,. .., and to do this we minimize the variational energy... 

Here the a, are the LCAO coefficients, which have to be determined. The formulation of HF theory where we use the LCAO approximation is usually attributed to Roothaan (1951a). His formulation applies only to electronic configurations of the type 1/ 3,...,. Following the discussion of Chapter 5, the charge... 

There is some small print to the derivation the orbitals must not change during the ionization process. In other words, the orbitals for the cation produced must be the same as the orbitals for the parent molecule. Koopmans (1934) derived the result for an exact HF wavefunction in the numerical Hartree-Fock sense. It turns out that the result is also valid for wavefunctions calculated using the LCAO version of HF theory. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

To illustrate the CISD technique, consider dineon (Figure 11.9). HF theory cannot hope to give an accurate description of the dispersion interaction between two neon atoms, so an electron correlation treatment is vital. Here are the results for a separation of 300 pm. 

Table 11.1 shows an interesting point about CISD. The energy of the dineon pair at the arbitrarily large separation of 5000 pm is exactly twice the energy of two free atoms at the HF-LCAO level of theory, but this is not the case at the CISD level of theory. We say that HF theory scales correctly, whilst CISD does not. 

There are, however, many electronic states for which a linear combination of determinants is essential, but which cannot by treated using HF theory. 

I have dealt at length with the Hartree and the Hartree-Foek models. The father of this field, Sir Wilham Hartree, was eoneemed with the atomie problem where it is routinely possible to integrate numerieally the HF integro-differential equations in order to produee (numerieal) wavefunetions that eorrespond to the Hartree-Foek limit. For moleeular applieations the LCAO variant of HF theory assumes a dominant role beeause of the redueed symmetry of the problem. 

The orbitals and orbital energies produced by an atomic HF-Xa calculation differ in several ways from those produced by standard HF calculations. First of all, the Koopmans theorem is not valid and so the orbital energies do not give a direct estimate of the ionization energy. A key difference between standard HF and HF-Xa theories is the way we eoneeive the occupation number u. In standard HF theory, we deal with doubly oecupied, singly occupied and virtual orbitals for which v = 2, 1 and 0 respectively. In solid-state theory, it is eonventional to think about the oecupation number as a continuous variable that can take any value between 0 and 2. 

The strength of DFT is tiiat only the total density needs to be considered. In order to calculate the kinetic energy with sufficient accuracy, however, orbitals have to be reintroduced. Nevertheless, as discussed in Section 6.5, DFT has a computational cost which is similar to HF theory, with the possibility of providing more accurate (exact, in principle) results. 

In the uncorrelated limit, where the many-electron Fock operator replaces the full electronic Hamiltonian, familiar objects of HF theory are recovered as special cases. N) becomes a HF, determinantal wavefunction for N electrons and N 1) states become the frozen-orbital wavefunctions that are invoked in Koopmans s theorem. Poles equal canonical orbital energies and DOs are identical to canonical orbitals. 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

In principle, the deficiencies of HF theory can be overcome by so-called correlated wavefunction or post-HF methods. In the majority of the available methods, the wavefunction is expanded in terms of many Slater-determinants instead of just one. One systematic recipe to choose such determinants is to perform single-, double-, triple-, etc. substitutions of occupied HF orbitals by virtual orbitals. Pictorially speaking, the electron correlation is implemented in this way by allowing the electrons to jump out of the HF sea into the virtual space in order... 

The connection to HF theory has been accomplished in a rather ingenious way by Kohn and Sham (KS) by referring to a fictitious reference system of noninteracting electrons. Such a system is evidently exactly described by a single Slater determinant but, in the KS method, is constrained to share the same electron density with the real interacting system. It is then straightforward to show that the orbitals of the fictitious system fulfil equations that very much resemble the HF equations ... 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

There are several things known about the exact behavior of Vxc(r) and it should be noted that the presently used functionals violate many, if not most, of these conditions. Two of the most dramatic failures are (a) in HF theory, the exchange terms exactly cancel the self-interaction of electrons contained in the Coulomb term. In exact DFT, this must also be so, but in approximate DFT, there is a sizeable self-repulsion error (b) the correct KS potential must decay as 1/r for long distances but in approximate DFT it does not, and it decays much too quickly. As a consequence, weak interactions are not well described by DFT and orbital energies are much too high (5-6 eV) compared to the exact values. 

Nevertheless, DFT has been shown over the past two decades to be a fairly robust theory that can be implemented with high efficiency which almost always surpasses HF theory in accuracy. Very many chemical and spectroscopic problems have been successfully investigated with DFT. Many trends in experimental data can be successfully explained in a qualitative and often also quantitative way and therefore much insight arises from analyzing DFT results. Due to its favorable price/performance ratio, it dominates present day computational chemistry and it has dominated theoretical solid state physics for a long time even before DFT conquered chemistry. However, there are also known failures of DFT and in particular in spectroscopic applications one should be careful with putting unlimited trust in the results of DFT calculations. 

It is well known that, within the framework of the MO-LCAO-HF theory, the electron density p at a given point r can be expressed as... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Hamiltonian in the CSF basis. This contrasts to standard HF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond structures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

In Hartree-Fock (HF) theory, the energy of a given orbital is lower when the orbital is occupied than when it is vacant (due to the formal self-energy in the former case), and the degeneracy is broken. Thus, perturbative F1F expressions such as Eq. (2.7) often have wider numerical validity than would be anticipated in naive MO theory. 

MOPAC is a general-purpose semiempirical molecular orbital program for the study of chemical structures and reactions. It is available in desktop PC running Windows, Macintosh OS, and Unix-based workstation versions. It uses semiempirical quantum mechanical methods that are based on Hartree-Fock (HF) theory with some parameterized functions and empirically determined parameters replacing some sections of the complete HF treatment. The approximations in... 

An alternative approach was offered by Lee, Yang, and Parr [19], who derived a gradient-corrected correlation functional ( LYP ) from the second-order density matrix in HF theory. Together with PW91, this functional is currently the most widely used correlation functional for molecular calculations. 

For the G2 set of compounds (a standardized test set of small molecules) the mean error to the atomization energy is approximately 2.5 kcal mol-1 at the B3LYP level, compared with 78 kcal -mol-1 for HF theory, and in the range of lkcalmor1 for the most accurate correlated ah initio methods. For most cases in which a moderately sized systems (10-50 atoms) is to be investigated, the B3LYP functional is currently the method of choice. 

The most usual starting point for approximate solutions to the electronic Schrodinger equation is to make the orbital approximation. In Hartree-Fock (HF) theory the many-electron wavefunction is taken to be the antisymmetrized product of one-electron wavefunctions (spin-orbitals) ... 

Equation (96) shows that the effective KS potential may be simply obtained by adding to the standard KS potential of the isolated solute, an electrostatic correction which turns out to be the RE potential Or, and the exchange- correlation correction 8vxc. It is worth mentioning here, that Eq (96) is formally equivalent to the effective Fock operator correction bfteffi defined in the context of the self consistent reaction field (SCRF) theory [2,3,14] within the HF theory, the exchange contribution is exactly self-contained in Or, whereas correlation effects are completely neglected. As a result, within the HF theory 8v = Or, as expected. 

Recently, quantum chemical computational techniques, such as density functional theory (DFT), have been used to study the electrode interface. Other methods ab initio methods based on Hartree-Fock (HF) theory,65 such as Mollcr-PIcsset perturbation theory,66,67 have also been used. However, DFT is much more computationally efficient than HF methods and sufficiently accurate for many applications. Use of highly accurate configuration interaction (Cl) and coupled cluster (CC) methods is prohibited by their immense computational requirements.68 Advances in computing capabilities and the availability of commercial software packages have resulted in widespread application of DFT to catalysis. 

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