Electronic structure problems

Applying Flartree-Fock wavefiinctions to condensed matter systems is not routine. The resulting Flartree-Fock equations are usually too complex to be solved for extended systems. It has been argried drat many-body wavefunction approaches to the condensed matter or large molecular systems do not represent a reasonable approach to the electronic structure problem of extended systems. 

The periodic nature of crystalline matter can be utilized to construct wavefunctions which reflect the translational synnnetry. Wavefiinctions so constructed are called Bloch functions [1]. These fiinctions greatly simplify the electronic structure problem and are applicable to any periodic system. 

Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree-Fock [68] wave functions to heat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the elech onic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69-71] restricted themselves to adiabatic problems, the method can incorporate non-adiabatic effects directly in the electionic wave function. 

At the present time, the solution of the electronic structure problem using full four component wave functions is far from routine [38]. In the future, as progress is made in this area, extension of the present approach to full four component wave functions can be expected. 

This algorithm alternates between the electronic structure problem and the nuclear motion It turns out that to generate an accurate nuclear trajectory using this decoupled algoritlun th electrons must be fuUy relaxed to the ground state at each iteration, in contrast to Ihe Car-Pairinello approach, where some error is tolerated. This need for very accurate basis se coefficients means that the minimum in the space of the coefficients must be located ver accurately, which can be computationally very expensive. However, conjugate gradient rninimisation is found to be an effective way to find this minimum, especially if informatioi from previous steps is incorporated [Payne et cd. 1992]. This reduces the number of minimi sation steps required to locate accurately the best set of basis set coefficients. 

Thereby the solution of the electronic-structure problem for an N-atomic system is decomposed into N locally self-consistent problems including only the M atoms in the LIZ associated with each atom in the system, and the computational effort now scales linearly with N, i.e. exhibits 0 N) scaling. 

In an ab initio simulation, the electronic structure problem is solved for each nuclear configuration, and forces are computed using the Hellmann-Feynman theorem. 

The actual form of the Hamiltonian operator hp does not have to be defined at this moment. As in standard perturbation theory, it is assumed that the solution of the electronic structure problem of the combined Hamiltonian HKS +HP can be described as the solution y/(0) of HKS, corrected by a small additional linear-response wavefunction /b//(,). Only these response orbitals will explicitly depend on time - they will follow the oscillations of the external perturbation and adopt its time dependency. Thus, the following Ansatz is made for the solution of the perturbed Hamiltonian HKS +HP ... 

In order to solve the electronic structure problem for a single geometry, the energy should be minimized with respect to the coefficients (see Eq. (5)) subject to the orthogonality constraints. This leads to the eigenvalue equation ... 

The first calculations on a two-electron bond was undertaken by Heitler and London for the H2 molecule and led to what is known as the valence bond approach. While the valence bond approach gained general acceptance in the chemical community, Robert S. Mulliken and others developed the molecular orbital approach for solving the electronic structure problem for molecules. The molecular orbital approach for molecules is the analogue of the atomic orbital approach for atoms. Each electron is subject to the electric field created by the nuclei plus that of the other electrons. Thus, one was led to a Hartree-Fock approach for molecules just as one had been for atoms. The molecular orbitals were written as linear combinations of atomic orbitals (i.e. hydrogen atom type atomic orbitals). The integrals that needed to be calculated presented great difficulty and the computations needed were... 

By replacing the wavefunction with a density matrix, the electronic structure problem is reduced in size to that for a two- or three-electron system. Rather than solve the Schrodinger equation to determine the wavefunction, the lower bound method is invoked to determine the density matrix this requires adjusting parameters so that the energy content of the density matrix is minimized. More precisely, the lower bound method requires finding a solution to the energy problem,... 

As presented, the Roothaan SCF process is carried out in a fully ab initio manner in that all one- and two-electron integrals are computed in terms of the specified basis set no experimental data or other input is employed. As described in Appendix F, it is possible to introduce approximations to the coulomb and exchange integrals entering into the Fock matrix elements that permit many of the requisite F, v elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level coulomb interaction integrals that can be computed in an ab initio manner. This approach forms the basis of so-called semi-empirical methods. Appendix F provides the reader with a brief introduction to such approaches to the electronic structure problem and deals in some detail with the well known Htickel and CNDO- level approximations. 

A central problem in physics and chemistry has always been the solution of the Schrodinger equation (SE) for stationary states. Such stationary states may relate to electronic structure problems, in which case one is primarily interested in bound states, or to scattering problems, in which case the stationary solutions are continuum states. In both cases, one of the most powerful tools in the theoretical arsenal for solving such problems is the partitioning technique (PT), which has been developed in a series of papers prominently by Per-Olov Lowdin [1-6] and Herman Feshbach [7-9]. 

Solve the electronic structure problem at the current positions. 

Finally, the use of ECP basis sets for heavy elements improves efficiency by reducing the scale of the electronic structure problem. In addition, relativistic effects can be accounted for by construction of the pseudopotential. 

In the preceding chapter we showed a number of electronic structure problems, where a proper wave function had to be constructed as a linear combination of several electronic configurations. In this chapter we will discuss the technical aspects in constructing these MCSCF wave functions and in determining the variational parameters - the Cl coefficients and the molecular orbitals. 

The emphasis will be more on understanding than on the technical aspects of the methods, and much time will be devoted to discussion of different electronic structure problems and the choice of appropriate methods for their solution. The course will consist of both lectures and exercises. 

As with the solution of other many-body electronic structure problems, determination of the unperturbed eigenvalues is numerically challenging and involves compromises in the following areas (1) approximations to the hamiltonian to simplify the problem (e.g., use of semi-empirical molecular orbital methods) (2) use of incomplete basis sets (3) neglect of highly excited states (4) neglect of screening effects due to other molecules in the condensed phase. 

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