Hartree-Fock theory

Ab initio Hartree-Fock theory is based on one single approximation, namely, the N-dectron wavefunction, F is restricted to an antisymmetrized product, a Slater determinant, of one-electron wavefunction, so called spin orbitals, 

Using this wavefunction the total energy of the electron system is minimized with respect to the choice of spin-orbitals under the constraint that the spin orbitals are orthogonal. The variational procedure is applied to this minimization problem and the result is the so called Hartree-Fock equations  

This is the single-electron operator including the electron kinetic energy and the potential energy for attraction to the nuclei (for convenience, the single electron is indexed as electron one). The two-electron operators in eq. (2.4) are defined as the Coulomb, J 

The spin orbitals can be separated into a spatial part, if/ (orbital or molecular orbital) and a spin eigenfunction, a (or for the opposite spin). In restricted Hartee-Fock theory (RHF), the spatial part is independent on the spin state, in contrast to unrestricted Hartree-Fock (UHF) where it is spin dependent. Consequently, the RHF spin orbitals can be written as %i= /6a whereas in UHF the corresponding relation is Xi= /, . The discussion here is limited to RHF case and to the case of an even number of electrons (closed shell system), but can easily be extended to treat also UHF1. 

By integrating out the spin-coordinate of eq. (2.6) it simplifies to the following RHF equation  

This operator is consistent with the leading terms in quasiparticle self-energies implied by many-body theory [275, 407]. 

The theory is based on an optimized reference state t that is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions / (r). This is the simplest form of the more general orbital functional theory (OFT) for an iV-electron system. The energy functional E = (4 // )is required to be stationary, subject to the orbital orthonormality constraint (i j) = Sij, imposed by introducing a matrix of Lagrange multipliers kj,. The general OEL equations derived above reduce to the UHF equations if correlation energy Ec and the implied correlation potential vc are omitted. The effective Hamiltonian operator is 

The theory is usually expressed in terms of canonical Hartree-Fock equations 

Janak s theorem, valid for general OEL equations when occupation numbers are varied, holds for the UHF theory in the form 

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)  

Here 21 is the antisymmetrizer which ensures that the Pauli Principle is obeyed. A convenient way of writing eq. (11.36) is as a Slater determinant  

The spin-orbitals are the products of spatial and spin factors, i.e. 

Given the trial wavefunction - the Slater determinant eq. (11.37) - we then use the variational principle to minimize the energy - the expectation value of the Hamiltonian H - with respect to the orbital coefficients cy (eq. (11.39)). This leads after a fair amount of algebra to the self-consistent Hartree-Fock equations  

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

Undoubtedly, the methods most widely used to solve the Schrodinger equation are those based on the approach originally proposed by Hartree [1] and Fock [2]. Hartree-Fock (HF) theory is the simplest of the ab initio or first principles quantum chemical theories, which are obtained directly from the Schrodinger equation without incorporating any empirical considerations. In the HF approximation, the n-electron wavefunction is built from a set of n independent one-electron spin orbitals which contain both spatial and spin components. The HF trial wavefunction is taken as a single Slater determinant of spin orbitals. 

The energy and spin orbitals are then determined variationally, subject to the constraint that the spin orbitals are orthonormal. This leads to the familiar HF integro-differential equations for the best one-electron orbitals. Physically, the HF approximation amounts to treating the individual electrons in the average field due to all the other electrons in the system. This effective Hamiltonian is called the Fock operator. 

However, the HF equations are still too difficult to solve in general, and so most commonly the molecular orbitals v/ (the spatial parts of the spin orbitals) are expanded as a linear combination of a finite set of atomic orbital basis functions ), i.e. 

The development of correlation methods which are both accurate and economical has been, and continues to be, one of the thorniest problems in modern quantum chemistry. There are a number of traditional approaches to the correlation problem. The correlation energy represents a small fraction of the total energy (usually less than one percent) and so Moller and Plesset proposed [6] that it be calculated via a perturbation technique. Another prominent class of correlation methods is configuration interaction (Cl) methods, which involve the variational addition to the wavefunction of substituted configurations, obtained by replacing occupied orbitals in the HF determinant with unoccupied (virtual) orbitals, i.e. 

The major strength of HF is that it is relatively economical, but across the spectrum of chemically interesting problems its ability as a predictive tool is only qualitative at best. We would like to be able to retain the inexpense of HF while enjoying the quality of the more sophisticated correlated methods. Of course, this seems like asking for too much, as it would appear to contradict the sage wisdom You get what you pay for . This may or may not be true, but in any case, it is clear that if the objective is to have a quantitatively accurate, cost-effective and widely applicable quantum mechanical method for predicting molecular energies, a radical departure from traditional correlation techniques will be necessary. 

The term that prevents an exact solution of the BO eigenvalue problem is the electron-electron repulsion that couples the motion of pairs of electrons. Without 

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 practice, unfortunately, not even the HF equations can be solved precisely due to the complicated shapes that the orbitals assume for general low-symmetry molecular environments. Hence, one introduces a fixed basis set ip that is used to expand the HF orbitals  

If the basis set is mathematically complete, then the equation holds precisely. In practice, one has to work with an incomplete finite basis set and hence the equality is only approximate. Results close to the basis set limit (the exact HF solutions) can nowadays be found, but for all practical intents and purposes, one needs to live with a basis set incompleteness error that must be investigated numerically for specific applications. 

The benefit is now that the HF equations are turned from complicated integro-differential equations into pseudo-eigenvalue equations for the unknown expansion coefficients c. 

In this chapter we will look at a few commonly used semiempirical methods and see how they evolved. Recently a comprehensive summary of the status of semiempirical methods, with the emphasis on MNDO, has been published. Other interesting reviews can be found in the book of Pople and Beveridge and that of Sadlej. In addition, reference is made to the review of AMI and PM3 that has been included as a chapter in Volume 1 of this series.  

We start our derivation of the Fock equations with the stationary state Schrodinger equation, 

The orbital approximation itself suggests that the many-electron wave-function v / can be written as a product of one-electron functions, (i), called orbitals.  

A is the antisymmetrizer, ensuring that the wavefunction changes sign on interchange of two electrons (and thus the wavefunction obeys the Pauli exclusion principle), and 0(S) is a spin projection operator that ensures that the wavefunction remains an eigenfunction of the spin-squared operator, 

For the development below, we will assume a closed-shell situation, with all electrons paired in molecular orbitals. In such a case OfSj = 1. In very many cases, however, an unrestricted Hartree-Fock (UHF) scheme is utilized for ground state properties. This theory is reasonably accurate for those cases in which each open-shell orbital has an electron of the same spin, i.e., the case that an open-shell has maximum multiplicity. In the UHF scheme Eq. [4] does not hold. Two Fock equations result, one for a and one for 3 spin molecular orbitals. In cases in which excited state properties are required, Eq. [4] is forced to hold in order to yield spectroscopic states, of known multiplicity. OfSJ can then become quite complex, and affects the form of the Fock operators that follow.  

In a few cases explicit expressions of determinant expansion coefficients or other matrix elements are given in this chapter in terms of two electron repulsion integrals over spatial orbitals in the so-called Mulliken or chemical notation 

All the ab initio methods discussed here are based on the Hartree-Fock (HF) or self-consistent field method. In closed-shell HF theory the unperturbed many-electron wavefunction is approximated by a single Slater determinant 

The spatial orbitals are solutions to the Hartree-Fock equations 

The Hartree-Fock Hamiltonian F is the sum of the one-electron Fock operators 

The N spin orbitals with the lowest energy, or N/2 spatial orbitals, are then used to construct the Af-electron Slater determinant i - he Hartree-Fock wave- 

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Mciny of the theories used in molecular modelling involve multiple integrals. Examples include tire two-electron integrals formd in Hartree-Fock theory, and the integral over the piriitii >ns and momenta used to define the partition function, Q. In fact, most of the multiple integrals that have to be evaluated are double integrals. 

Linear Combination of Atomic Orbitals (LCAO) in Hartree-Fock Theory... 

Several functional forms have been investigated for the basis functions Given the vast experience of using Gaussian functions in Hartree-Fock theory it will come as no surprise to learn that such functions have also been employed in density functional theory. However, these are not the only possibility Slater type orbitals are also used, as are numerical... 

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

Recently, a third class of electronic structure methods have come into wide use density functional methods. These DFT methods are similar to ab initio methods in many ways. DFT calculations require about the same amount of computation resources as Hartree-Fock theory, the least expensive ab initio method. 

Reproducing the exact solution for the relevant n-electron problem a method ought to yield the same results as the exact solution to the Schrodinger equation to the greatest extent possible. What this means specifically depends on the theory underlying the method. Thus, Hartree-Fock theory should be (and is) able to reproduce the exact solution to the one electron problem, meaning it should be able to treat cases like HeH ... 

The first cell in the last tow of the table represents the Hartree-Fock limit the best approximation that can be achieved without taking electron correlation into account. Its location on the chart is rather far from the exact solution. Although in some cases, quite good results can be achieved with Hartree-Fock theory alone, in many others, its performance ranges from orfly fair to quite poor. We ll look at some these cases in Chapters 5 and 6. 

Diffuse functions have very little effect on the optimized structure of methanol but do significantly affect the bond angles in negatively charged methoxide anion. We can conclude that they are required to produce an accurate structure for the anion by comparing the two calculated geometries to that predicted by Hartree-Fock theory at a very large basis set (which should eliminate basis set effects). 

As we have seen throughout this book, the Hartree-Fock method provides a reasonable model for a wide range of problems and molecular systems. However, Hartree-Fock theory also has limitations. They arise principally from the fact that Hartree-Fock theory does not include a full treatment of the effects of electron correlation the energy contributions arising from electrons interacting with one another. For systems and situations where such effects are important, Hartree-Fock results may not be satisfactory. The theory and methodology underlying electron correlation is discussed in Appendix A. 

Hartree-Fock theory is very useful for providing initial, first-level predictions for many systems. It is also reasonably good at computing the structures and vibrational frequencies of stable molecules and some transition states. As such, it is a good base-level theory. However, its neglect of electron correlation makes it unsuitable for some purposes. For example, it is insufficient for accurate modeling of the energetics of reactions and bond dissociation. 

When we consider the predicted atomization energy, however, we see vast differences among the functionals. Like Hartree-Fock theory, the SVWN and SVWN5 functionals are completely inadequate for predicting this system s atomization energy (which is not an atypical result). The BLYP value is also quite poor. 

The DFT and MP2 calculations produce very similar structures, although the BLYP bond length is again longer than those of the other functionals. Hartree-Fock theory predicts a bond length which is significantly shorter than the methods including electron correlation. 

These SVWN5 results are somewhat fortuitous. Be careful not to overgeneralize from their agreement to experiment. We will see a different result in Exercise 6.7. Several other excerises will also include comparisons of DFT methods to Hartree-Fock theory, MP2 and other electron correlation methods. 

Hartree-Fock theory does quite a poor job of predicting the structures and frequencies for these compounds. It produces highly distorted structures in all three cases, and its computed frequencies bear little resemblance to the experimental observations. MP2 theory generally does better for the structures, although it fails to located a distorted structure for Na F3. The frequencies computed at the MP2 level also vary widely from experiment. 

The UHF curve is much higher than those for the correlation methods Hartree-Fock theory does a relatively poor job of describing this process. The MP2 curve is somewhat higher than those for the MP3 and MP4(SDTQ) levels, which appear to have converged. 

The Hartree-Fock values range firom good to quite poor. For the first reaction, cancellation of errors allows Hartree-Fock theory to predict a good value for AH (it overestimates the energies for both ethane and acetone, and underestimates the one for acetaldehyde). 

AMI benefits from the same cancellation of errors for the first reaaion as Hartree-Fock theory. However, it performs even more poorly for the other two reactions. ... 

Like Hartree-Fock theory, Cl-Singles is an inexpensive method that can be applied to large systems. When paired with a basis set, it also may be used to define excited state model chemistries whose results may be compared across the full range of practical systems. 

Despite these comparisons to Hartree-Fock theory, the O-Singles method does include some electron correlation. 

Note that the frequency calculation produces many more frequencies than those listed here. We ve matched calculated frequenices to experimental frequencies using symmetry types and analyzing the normal mode displacements. The agreement with experiment is generally good, and follows what might be expected of Hartree-Fock theory in the ground state. ... 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

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. 

Any method which goes beyond SCF in attempting to treat this phenomenon properly is known as an electron correlation method (despite the fact that Hartree-Fock theory does include some correlation effects) or a post-SCT method. We will look briefly at two different approaches to the electron correlation problem in this section. 

Configuration Interaction (Cl) methods begin by noting that the exact wavefunction 4 cannot be expressed as a single determinant, as Hartree-Fock theory assumes. Cl proceeds by constructing other determinants by replacing one or more occupied orbitals within the Hartree-Fock determinant with a virtual orbital. 

Another approach to electron correlation is Moller-Plesset perturbation theory. Qualitatively, Moller-Plesset perturbation theory adds higher excitations to Hartree-Fock theory as a non-iterative correction, drawing upon techniques from the area of mathematical physics known as many body perturbation theory. 

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