The Roothaan equations

The preceding discussion means that the Matrix equations already described are correct, except that the Fock matrix, F, replaces the effective one-electron Hamiltonian matrix, and that Fdepends on the solution C  

These are the Roothaan SCFequations, which clearly can be solved iteratively—guess C, form F, diagonalize to a new C, form a new F. and so on. 

If you define a density matrix P by summing over all occupied molecular orbitals  

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HyperChcin s ah mitio calculations solve the Roothaan equations (.h9 i on page 225 without any further approximation apart from th e 11 se of a specific fin iie basis set. Th ere fore, ah initio calcii lation s are generally more accurate than semi-enipirical calculations. They certainly involve a more fundamental approach to solving the Sch riidiiiger ec nation than do semi-cmpineal methods. 

Since the first formulation of the MO-LCAO finite basis approach to molecular Ilartree-Pock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — arc calculated and stored on external storage. Th e second step then con sists of the iterative solution of the Roothaan equations, where the integrals from the first step arc read once for every iteration. 

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. 

The Roothaan equations just described are strictly the equations for a closed-shell Restricted Hartree-Fock (RHF) description only, as illustrated by the orbital energy level diagram shown earlier. To be more specific ... 

The Roothaan equations are the basic equations for closed-shell RHF molecular orbitals, and the Pople-Nesbet equations are the basic equations for open-shell UHF molecular orbitals. The Pople-Nesbet equations are essentially just the generalization of the Roothaan equations to the case where the spatials /j and /jP, as shown previously, are not defined to be identical but are solved independently. 

Gianinetti, E., Raimondi, M. and Tomaghi, E. (1996) Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations, Int. J. Quantum Chem., 60, 157-166. 

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. 

The proper way of dealing with periodic systems, like crystals, is to periodicize the orbital representation of the system. Thanks to a periodic exponential prefactor, an atomic orbital becomes a periodic multicenter entity and the Roothaan equations for the molecular orbital procedure are solved over this periodic basis. Apart from an exponential rise in mathematical complexity and in computing times, the conceptual basis of the method is not difficult to grasp [43]. Software for performing such calculations is quite easily available to academic scientists (see, e.g., CASTEP at www.castep.org CRYSTAL at www.crystal.unito.it WIEN2k at www.wien2k.at). 

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. 

Despite these shortcomings, the Roothaan equation has been used extensively and the Hartree-Fock energies of various small molecules have been calculated. However, the difficulties encountered in calculating the energy of large molecules are such that simplified methods are desirable in these cases. Several such methods will be discussed in the next section ... 

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. 

Thus, the practical way of solving the Roothaan equations is to choose an initial set of cr° s, calculate the F s, solve the equations for a new set, and iterate again until consistency is attained. 

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... 

Then the Roothaan equations can be written in the matrix form ... 

Solution of the Roothaan equations calls for laborious computations which become more and more so as the basis broadens, i.e. the number of AOs increases. In the last two decades theoreticians have invented a great number of clever tricks in their attempts to find how to calculate molecules the Roothaan way . However, the system of equations (4) turned out to be very tough to handle. Despite the ingenuity of researchers and the advancement in computer technique practical application of the Roothaan method is substantially limited by the size of molecular systems. 

These are known as the Roothaan equations. They represent an algebraic equivalent to the Hartree-Fock equations. The approximate eigenvalues represent orbital energies. By Koopmans theorem, — approximates the ionization energy for an electron occupying orbital a. The orbital energies can be determined directly Ifom the n roots of the secular equation... 

Tgj is represented exactly and the exact electronic energy, which also includes dispersion effects correctly, is obtained. However, this comes with infinite computational costs. Hence, methods needed to be devised, which allow us to approximate the infinite expansion in Eq. (12.9) by a finite series to be as short as possible. A straightforward approach is the employment of truncated configuration interaction (CI) expansions. Note that (electronic) configuration refers to the set of molecular orbitals used to construct the corresponding Slater determinant. It is a helpful notation for the construction of the truncated series in a systematic manner and yields a classification scheme of Slater determinants with respect to their degree of excitation . Excitation does not mean physical excitation of the molecule but merely substitution of orbitals occupied in the Hartree-Eock determinant o by virtual, unoccupied orbitals. Within the LCAO representation of molecular orbitals the virtual orbitals are obtained automatically with the solution of the Roothaan equations for the occupied orbitals that enter the Hartree-Eock determinant. 

By substituting W with Eq. (5), integrating, and applying the variation principle, the Schrodinger equation (1) is converted into a system of linear equations, the Roothaan equation [12] (recently reviewed by Zerner [13]). 

Since the Fock matrix is dependent on the orbital coefficients, the Roothaan equations have to be repeatedly solved in an iterative process, the self-consistent field (SCF) procedure. One important step in the SCF procedure is the conversion of the general eigenvalue equation (7) into an ordinary one by an orthogonalization transformation... 

The PPP method is the first semiempirical method presented here where the Fock matrix does depend on the MO coefficients C [via the density matrix elements P, see Eq. (11)]. Therefore, the Roothaan equations (by definition due to the ZDO approximation) in the orthogonal basis, Eq. (13), have to be solved in an iterative process until self-convergence is achieved [self-consistent field (SCF) procedure]. As starting coefficients C°, usually the orbitals of an HMO calculation are used. 

A many-electron wave function can be constructed from the set of occupied one-electron spinorbitals in the form of the Slater determinant. The molecular orbitals in the LCAO form are determined by solving the Roothaan equations. The MO method is improved by the configuration interaction. For evaluation of the matrix elements of operators over the determinantal functions, the Slater rules are helpful. 

Saebo, S., Tong, W. and Pulay, P. Efficient elimination of basis set superposition errors by the local correlation method Accurate ab initio studies of the water dimer,/. Chem. Phys., 98, 2170-2175. Gianinetti, E., Raimondi, M. and Tomaghi, E. (1996) Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations, Int. J. Quantum Chem., 60, 157-166. Eamulari, A., Raimondi, M., Sironi, M. and Gianinetti, E. (1998) Hartree-Focklimit properties of water dimer in absence of BSSE, Chem. Phys., 232, 275-287. 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

One starts with guesses for the occupied-MO expressions as linear combinations of the basis functions, as in (13.156). This initial set of MOs is used to compute the Fock operator F from (13.149) to (13.152). The matrix elements (13.158) are computed, and the secular equation (13.159) is solved to give an initial set of c, s.These c, s are used to solve (13.157) for an improved set of coefficients, giving an improved set of MOs, which are then used to compute an improved F, and so on. One continues until no further improvement in MO coefficients and energies occurs from one cycle to the next. The calculations are done using a computer. (The most efficient way to solve the Roothaan equations is to use matrix-algebra methods see the last part of this section.)... 

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... 

The Fock Matrix Elements. To solve the Roothaan equations (13.157), we first must express the Fock matrix elements (integrals) in terms of the basis functions x-The Fock operator F is given by (13.149), and... 

To solve the Roothaan equations (13.157), we need the integrals F and S,. The overlap integrals are... 

Substitution of the lower root cj into the Roothaan equation (13.157) with r = 2 gives... 

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