The Hartree-Fock-Roothaan SCF Equation

Note in particular that we have to integrate over the spin functions as well as the spatial coordinates using simple rules  

Much of the mathematics for the SCF process was derived for atoms using texts like Condon and Shortly [5], but in 1951, Roothaan [1] solved the problem for molecules using the LCAO philosophy. Instead of using just single orbitals such as l a in our example above for the Li atom, the LCAO concept says that (1,2. ri) is built from many spin orbitals t t,- that are linear combinations of basis functions Thus, vlr = The original paper by Roothaan [1] is 

After collecting aU the nonzero terms in the calculation of ( ) = we find for just the 

In the grand scheme of the detailed derivation, you will be looking at energy terms like 

As a result of spin orthogonality, only (1 /2) of the exchange terms are nonzero but they are there This was first pointed out by Fock [6] and was added as a correction to the method then developed by Hartree [7]. The combined method is now called the Hartree-Fock method if tabulated numerical orbitals are used, but the Hartree-Fock-Roothaan method in an LCAO basis. Today this method is 

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The Hartree-Fock-Roothaan SCF equations, expressed in terms of the matrix elements of the Fock operator Frs, and the overlap matrix elements Srs, take the form ... 

If the basis set used is finite and incomplete, solution of the secular equation yields approximate, rather than exact, eigenvalues. An example is the linear variation method note that (2.78) and (1.190) have the same form, except that (1.190) uses an incomplete basis set. An important application of the linear variation method is the Hartree-Fock-Roothaan secular equation (1.298) here, basis AOs centered on different nuclei are nonorthogonal. Ab initio and semiempirical SCF methods use matrix-diagonalization procedures to solve the Roothaan equations. 

We shall consider first the solution of the Hartree-Fock (HF) SCF equations. The molecular orbitals are typically expanded in an atomic basis set of Gaussian functions, which leads to the Roothaan-Hall equations for the orbital coefficients C as expressed in the atomic orbital space. 

We now consider the PPP, CNDO, INDO, and MINDO two-electron semiempirical methods. These are all SCF methods which iteratively solve the Hartree-Fock-Roothaan equations (1.296) and (1.298) until self-consistent MOs are obtained. However, instead of the true Hartree-Fock operator (1.291), they use a Hartree-Fock operator in which the sum in (1.291) goes over only the valence MOs. Thus, besides the terms in (1.292), f/corc(l) m these methods also includes the potential energy of interaction of valence electron 1 with the field of the inner-shell electrons rather than attempting a direct calculation of this interaction, the integrals of //corc(/) are given by various semiempirical schemes that make use of experimental data furthermore, many of the electron repulsion integrals are neglected, so as to simplify the calculation. 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

Quantum mechanical calculations are carried out using the Variational theorem and the Har-tree-Fock-Roothaan equations.t - Solution of the Hartree-Fock-Roothaan equations must be carried out in an iterative fashion. This procedure has been called self-consistent field (SCF) theory, because each electron is calculated as interacting with a general field of all the other electrons. This process underestimates the electron correlation. In nature, electronic motion is correlated such that electrons avoid one another. There are perturbation procedures whereby one may carry out post-Hartree-Fock calculations to take electron correlation effects into account. " It is generally agreed that electron correlation gives more accurate results, particularly in terms of energy. 

Differentiation of the basis set form of SCF, the Hartree-Fock-Roothaan equation, is complicated by the fact that the Fock operator is itself dependent on the orbital set. But this complication is not difficult to deal with. We will use C to represent the matrix of orbital expansion coefficients, S to be the matrix (operator) of the overlap of basis functions, F to be the Fock operator matrix, and E to be the orbital eigenvalue (orbital energy) matrix. The equation to be differentiated is... 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

The most simple approach is the Hartree-Fock (HF) self-consistent field (SCF) approximation, in which the electronic wave function is expressed as an antisymmetrized product of one-electron functions. In this way, each electron is assumed to move in the average field of all other electrons. The one-electron functions, or spin orbitals, are taken as a product of a spatial function (molecular orbital) and a spin function. Molecular orbitals are constructed as a linear combination of atomic basis functions. The coefficients of this linear combination are obtained by solving iteratively the Roothaan equations. 

Roothaan equations have been modified in a previous work with the aim of avoiding BSSE at the Hartree-Fock level of theory. The resulting scheme, called SCF-MI (Self Consistent Field for Molecular Interactions), underlines its special usefulness for the computation of intermolecular interactions. 

This representation permits analytic calculations, as opposed to fiiUy numerical solutions [47,48] of the Hartree-Fock equation. Variational SCF methods using finite expansions [Eq. (2.14)] yield optimal analytic Hartree-Fock-Roothaan orbitals, and their corresponding eigenvalues, within the subspace spanned by the finite set of basis functions. 

In spite of these gross approximations, the method proved to be extremely useful and was extensively used to correlate the chemical properties of conjugated systems. Several attempts were subsequently made to introduce the repulsions between the n electrons in the calculations. These include the work of Goeppert-Mayer and Sklar 4> on benzene and that of Wheland and Mann 5> and of Streitwieser 6> with the a> technique. But the first general methods of wide application were developed only in 1953 by Pariser and Parr 7> (interaction of configuration) and by Pople 8> (SCF) following the publication by Roothaan of his self-consistent field formalism for solving the Hartree-Fock equation for... 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

In this review, research in the field of van der Waals molecules accomplished by our group in the last few years was summarised. On the basis of the results obtained so far, it appears that the modification of the Roothaan equations to avoid basis set superposition error at the Hartree-Fock level of theory is a promising approach. The fundamental development of the SCF-MI strategy to deal with electron correlation treatments in the framework of the valence bond theory has been described. A compact multistructure and size... 

We have used the terms SCF wave function and Hartree-Fock wave function interchangeably. In practice, the term SCF wave function is applied to any wave function obtained by iterative solution of the Roothaan equations, whether or not the basis set is large enough to give a really accurate approximation to the Hartree-Fock SCF wave function. There is only one true Hartree-Fock SCF wave function, which is the best possible wave function that can be written as a Slater determinant of spin-orbitals. Some of the extended-basis-set calculations approach the true Hartree-Fock wave... 

Kim has formulated a relativistic Hartree-Fock-Roothaan equation for the ground states of closed-shell atoms using Slater-type orbitals. Relativistic effects in atoms have been reviewed by Grant. Malli and coworkers have formulated a relativistic SCF method for molecules. In this method, four-component spinor wavefunctions are obtained variationally in a self-consistent scheme using Gaussian basis sets. 

In order to determine these unknowns the variational minimax principle of chapter 8 is invoked. For this procedure, we may again start from the energy expression of section 10.2 and differentiate it or directly insert the basis set expansion of Eq. (10.3) into the SCF Eqs. (8.185). These options are depicted in Figure 10.2. The resulting Dirac-Hartree-Fock equations in basis set representation are called Dirac-Hartree-Fock-Roothaan equations according to the work by Roothaan [511] and Hall [512] on the nonrelativistic analog. 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

Fock modified Hartree s SCF method to include antisymmetrization. Roothaan further modified the Hartree-Fock method by representing the orbitals by linear combinations of basis functions similar to Eq. (16.3-34) instead of by tables of numerical values. In Roothaan s method the integrodifferential equations are replaced by simultaneous algebraic equations for the expansion coefficients. There are many integrals in these equations, but the integrands contain only the basis functions, so the integrals can be calculated numerically. The calculations are evaluated numerically. This work is very tedious and it is not practical to do it without a computer. 

We have now succeeded in expressing the Hartree-Fock equations in the AO basis, avoiding any transformation to the MO basis. The pseudo-eigenvalue equations (10.6.16) are called the Roothaan-Hall equations [3,4]. In Exercise 10.4, the Roothaan-Hall SCF procedure is used to calculate the Hartree-Fock wave function for HeH in the STO-3G basis. 

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