Fock-Roothaan equations

These O, are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5)), a simple set of equations (the Hartree-Fock-Roothaan equations) is obtained which can be used to determine the optimum coefficients Cti. For those systems where the space part of each MO is doubly occupied, i.e. there are two electrons in each 0, with spin a and spin respectively so that the complete MOs including spin are different, the total wavefunction is... 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

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. 

Thus four of the seven lowest H20 MOs are linear combinations of the four a, symmetry orbitals listed above, and are a, MOs similarly, the two lowest b2 MOs are linear combinations of 02p and H,1j — H21.s, and the lowest bx MO is (in this minimal-basis calculation) identical with 02px. The coefficients in the linear combinations and the orbital energies are found by iterative solution of the Hartree-Fock-Roothaan equations. One finds the ground-state electronic configuration of H20 to be... 

Now we have FC = SCe (5.57), the matrix form of the Roothaan-Hall equations. These equations are sometimes called the Hartree-Fock-Roothaan equations, and, often, the Roothaan equations, as Roothaan s exposition was the more detailed and addresses itself more clearly to a general treatment of molecules. Before showing how they are used to do ab initio calculations, a brief review of how we got these equations is in order. 

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. 

This is a secular equation whose roots give the orbital energies e,.The Hartree-Fock-) Roothaan equations (13.157) must be solved by an iterative process, since the F integrals depend on the orbitals <, (through the dependence of F on the <, s), which in turn depend on the unknown coefficients Cj,. 

Performing Kohn-Sham Density-Functional Calculations. How does one do a molecular density-functional calculation with (or some other functional) One starts with an initial guess for p, which is usually foimd by superposing calculated electron densities of the individual atoms at the chosen molecular geometry. From the initial guess for p(r), an initial estimate of u c( ) found from (15.127) and (15.131) and this initial v d ) is used in the Kohn-Sham equations (15.121), which are solved for the initial estimate of the KS orbitals. In solving 15.121), the flP s are usually expanded in terms of a set of basis functions Xr ( P = 2r=i to yield equations that resemble the Hartree-Fock-Roothaan equations (13.157) and (13.179), except that the Fock matrix elements = xr F x are replaced by the Kohn-Sham matrix elements = (Xr Xs), where is in (15.122) and(15.123).Thus, instead of (13.157), in KS DFT with a basis-set expansion of the orbitals, one solves the equations... 

When the Roothaan equations (13.157) [or (13.179)] are solved exactly, the canonical MOs and the calculated values of molecular properties do not change if one changes the orientation of the coordinate axes the calculated values are said to be rota-tionally invariant Likewise, the results do not change if each basis AO on a particular atom is replaced by a linear combination of the basis AOs on that atom, and the results are hybridizntioruilly invariant When approximations are made in solving the Hartree-Fock-Roothaan equations, rotational and hybridizational invariance may not hold. 

The secular equations for the Fock operator will have, of course, the form of the Hartree and Fock-Roothaan equations (cf. Chapter 8, p. 431) ... 

What has been said previously about the Hartree-Fock method is only a sort of general theory. The time has now arrived to show how the method works in practice. We have to solve the Haitree-Fock-Roothaan equation (ef. Chapter 8, pp. 431 and 531). 

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. 

Use of the LCAO expansion leads to the Hartree-Fock-Roothaan equations Fc = See. Our job is then to find the LCAO coefficients c. This is achieved by transforming the matrix equation to the form of the eigenvalue problem, and to diagonalize the corresponding Hermitian matrix. The canonical molecular orbitals obtained are linear combinations of the atomic orbitals. The lowest-energy orbitals are occupied by electrons, those of higher energy are called virtual and are left empty. 

We solve the Hartree-Fock-Roothaan equation (see Appendix L available at booksite. elsevier.com/978-0-444-59436-5, p. el07) and obtain the M MOs, we choose the N/2 occupied orbitals (those of the lowest energy). 

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. 

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... 

In this section, we examine the main modifications in the Hartree-Fock Roothaan equations (4.33), it being necessary to take into account the translation symmetry of periodic systems. The first most important difference appears in the LCAO representation of the crystalline crbitals (CO) compared to the molecular orbitals (MO). 

The exact solntion of Hartree-Fock-Roothaan equations (4.33) for molecular systems means use of a complete set of basis functions (such a solution corresponds to the Hartree-Fock hmit and in practice can be achieved mainly for the simple molecules). 

Derivation of the Relativistic Hartree-Fock-Roothaan Equations for Molecules and Crystals... 

In complete analogy to the derivation of the nonrelativistic Hartree-Fock-Roothaan equations/ the generalized overlap matrix S" is here given by... 

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