SCF calculation

This chapter is devoted to SCF calculations which represent the overwhelming majority of the reported ab initio calculations. We are not going here to treat general problems of SCF theory which are also inherent in popular semiempirical methods such as PPP and CNDO, These features will be intentionally suppressed. Instead, emphasis will be laid on specific problems of the ab initio SCF approach that are not encountered in semiempirical MO methods and on the progress achieved in solution of these problems in last years. 

As with semiempirical methods, the problem to be solved is given by the Hartree-Fock-Roothaan (HFR) equations 

LCAO expansion (2.1). For closed-shell systems it holds 

Generalization to open-shell systems does not represent any specific problem of ab initio calculations and therefore it will not be treated here. In eqn, (3.3), denotes the matrix element of the one-electron part of Hamiltonian, is the element of the density matrix 

In semiempirical methods of the PPP and CNDO types, the elements are approximated by expressions containing empirical parameters. In ab initio treatments they are of course calculated rigorously. Since the one-electron part of Hamiltonian (1.1) contains two terms, the elements are composed of two contributions 

A particular variant of the coupled cluster method, called Fock-space or valence-universal [49,50], gave remarkable agreement with experiment for many transition energies of heavy atoms [51]. This success makes the scheme a useful tool for reliable prediction of the structure and spectrum of superheavy elements, which are difficult to access experimentally. A brief description of the method is given below. A more flexible scheme with higher accuracy and extended applicability, the intermediate Hamiltonian Fock-space coupled cluster approach, is shown in the next section. 

The Dirac-Coulomb-Breit Hamiltonian may be rewritten in second-quantized form [5,15] in terms of normal-ordered products of spinor creation and annihilation operators r s and 

Here frs and (7 s tu) are, respectively, elements of one-electron Dirac-Fock-Breit and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators A+ is now taken over by normal ordering, denoted by the curly braces in (16), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [49,50], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some 

This partitioning allows for partial decoupling of the open-shell CC equations. The equations for the (m, n) sector involve only S elements from sectors (fc, 1) with k m and / n, so that the very large system of coupled nonlinear equations is separated into smaller subsystems, which are solved consecutively first, the equations for are iterated to convergence the 5(1,0) 

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In this approach [ ], the LCAO-MO coefficients are detemiined first via a smgle-configuration SCF calculation or an MCSCF calculation using a small number of CSFs. The Cj coefficients are subsequently detemiined by making the expectation value ( P // T ) / ( FIT ) stationary. 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Whereas it is generally sufficient (at least for the pubhshed methods) to specify the semi-empirical MO technique used in order to define the exact method used for the calculations, ab-initio theory offers far more variations, so that the exact level of the calculation must be specified. The starting point of most ab-initio jobs is an SCF calculation analogous to those discussed above for semi-empirical MO calculations. In ab-initio theory, however, all necessary integrals are calculated correctly, so that the calculations are very much (by a factor of about 1000) more time-consuming than their semi-empirical counterparts. 

Not all Iterative semi-empirical or ah iniiio calculations converge for all cases. For SCF calculation s of electronic stnictiire. system s with a small energy gap between the highest occupied orbital and the lowest unoccupied orbital may not converge or may converge slowly. (They are generally poorly described by the Ilartree-Foch method.)... 

The extent to which this condition does not occur is a m easiire of deviance from self-con sisten cy. Th e DIIS melh od ii ses a lin ear combination of previoii s Fock matrices to predict a Fock matrix that minimizes [I - K. This new Rich matrix is then used by the SCF calculation. 

III an SCF calculation. many iterations may beneetled to achieve SCr con vergeiice. In each iteration all the two-electron integrals are retrieved to form a Fock matrix. Fast algorith m s to retrieve the two-cicetron s integrals arc important. 

Living calculated the integrals, we are now ready to start the SCF calculation. To formulate the Fock mahix it is necessary to have an initial guess of the density matrix, P. The simplest approach is to use the null matrix in which all elements are zero. In this initial step the Fock nulrix F is therefore equal to... 

A Hbasis functions provides K molecular orbitals, but lUJiW of these will not be occupied by smy electrons they are the virtual spin orbitals. If u c were to add an electron to one of these virtual orbitals then this should provide a means of calculating the electron affinity of the system. Electron affinities predicted by Konpman s theorem are always positive when Hartree-Fock calculations are used, because fhe irtucil orbitals always have a positive energy. However, it is observed experimentally that many neutral molecules will accept an electron to form a stable anion and so have negative electron affinities. This can be understood if one realises that electron correlation uDiild be expected to add to the error due to the frozen orbital approximation, rather ihan to counteract it as for ionisation potentials. 

Rinaldi D, M F Ruiz-Lopez and J L Rivail 1983. Ab Initio SCF Calculations on Electrostatically Solvate Molecules Using a Deformable Three Axes Ellipsoidal Cavity. Journal of Chemical Physics 78 834 838. 

In PPP-SCF calculations, we make the Bom-Oppenheimer, a-rr separation, and single-electron approximations just as we did in Huckel theor y (see section on approximate solutions in Chapter 6) but we take into account mutual electrostatic repulsion of n electrons, which was not done in Huckel theory. We write the modified Schroedinger equation in a form similar to Eq. 6.2.6... 

HMO method. This is because electron repulsion is taken into account in the SCF calculation whereas it is not taken into account in the Huckel calculation. 

HEAT OF FORl-lATION SCF CALCULATIONS COMPUTATION TIME... 

In an SCF calculation, the energies from one iteration to the next can follow one of several patterns ... 

There are quite a number of ways to effectively change the equation in an SCF calculation. These include switching computation methods, using level shifting, and using forced convergence methods. 

If you have an SCF calculation that failed to converge, which of the techniques outlined here should you try first Here are our suggestions, with the preferred techniques listed first ... 

The original PCM method uses a cavity made of spherical regions around each atom. The isodensity PCM model (IPCM) uses a cavity that is defined by an isosurface of the electron density. This is defined iteratively by running SCF calculations with the cavity until a convergence is reached. The self-consistent isodensity PCM model (SCI-PCM) is similar to IPCM in theory, but different in implementation. SCI-PCM calculations embed the cavity calculation in the SCF procedure to account for coupling between the two parts of the calculation. 

For systems with unpaired electrons, it is not possible to use the RHF method as is. Often, an unrestricted SCF calculation (UHF) is performed. In an unrestricted calculation, there are two complete sets of orbitals one for the alpha electrons and one for the beta electrons. These two sets of orbitals use the same set of basis functions but different molecular orbital coefficients. 

The HE, GVB, local MP2, and DFT methods are available, as well as local, gradient-corrected, and hybrid density functionals. The GVB-RCI (restricted configuration interaction) method is available to give correlation and correct bond dissociation with a minimum amount of CPU time. There is also a GVB-DFT calculation available, which is a GVB-SCF calculation with a post-SCF DFT calculation. In addition, GVB-MP2 calculations are possible. Geometry optimizations can be performed with constraints. Both quasi-Newton and QST transition structure finding algorithms are available, as well as the SCRF solvation method. 

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