Calculating Electronic Excited States

Introduction

These same techniques may also be necessary to find the ground state wave function. Determining the ground state electron configuration can be particularly difficult for compounds with very low energy excited states.

This document is concerned only with ab initio calculations. These are calculations in which only the theory of quantum mechanics is applied with no experimental data used in the calculation. Some of these techniques are also applicable to certain classes of semiempirical calculations.

This write up is not intended to be a detailed description. It is meant to be a short introduction with a discussion of strengths and weaknesses of various methods. This should also serve as a check list of available methods when faced with computational difficulties. References are provided for more detailed discussions information.

Spin states

CIS

A configuration interaction calculation uses molecular orbitals that have been optimized typically with a Hartree-Fock (HF) calculation. Generalized Valence Bond (GVB) and multi-configuration self consistent field (MCSCF) calculations can also be used as a starting point for a configuration interaction calculation.

A CIS calculation starts with this initial set of orbitals and moves one electron to one of the virtual orbitals from the original calculation. This gives a description of one of the excited states of the molecule, but does not change the quality of the description of the ground state as double excitation CIs do. This gives a wave function of somewhat lesser quality than the original calculation since the orbitals have been optimized for the ground state. Often this results in the ground state energy being a bit low relative to the other states.

A CIS calculation is not extremely accurate. However, it has the advantage of being able to compute many excited state energies easily.

Initial guess

The advantage of this method is that the orbitals have been optimized for the excited state.

The disadvantage is that there is no guarantee that it will work. If there is no energy barrier between the initial guess and the ground state wave function, the entire calculation will converge back to the ground state. The convergence path may take the calculation to an undesired state in any case.

A second disadvantage of this technique applies if the state is the same symmetry as a lower energy state. There is no guarantee that the state obtained is completely orthogonal to the ground state. This means that the wave function obtained may be some mix of the lower energy state and a higher energy state. In practice, this type of calculation only converges to a higher state if a fairly reasonable description of the excited state wave function is obtained. Mixing tends to be a significant concern if the orbital energies are very close together or the system is very sensitive to correlation effects.

Block diagonal Hamiltonians

Some programs such as COLUMBUS, DMOL and GAMESS actually set up a separate matrix for each irriducible representation and solve them separately. Such programs give the user the option of defining how many electrons are of each irreducible representation. This defines the symmetry of the wave function. In this case the resulting wave function is the lowest energy wave function of a particular symmetry.

This is a very good way to get excited states which differ in symmetry from the ground state and are the lowest energy state within that symmetry.

Higher roots of a CI

This is a very reliable way to get a high quality wave function for the first few excited states. For higher excited states, CPU times become very large since more iterations are generally needed to converge the CI calculation.

Neglecting a basis function

This is a reliable way to get an excited state wave function even when it is not the lowest energy wave function of that symmetry. However it might take a bit of work to construct the input file depending upon the individual program.

Imposing orthogonality - DFT techniques

The easiest excited states to find using density functional theory techniques are those which are the lowest state of a given symmetry thus using a ground state calculation.

A promising techniques is one which uses a variational bound for the average of the first M states of a molecule.

A few other options have been examined. However, there is not yet a large enough volume of work applying DFT to excited states to predict the reliability of any of these techniques.

Imposing orthogonality - QMC techniques

There are a number of methods for getting excited state energies from QMC calculations. These methods will only be mentioned here and are explained more fully in the text by Hammond, Lester and Reynolds referenced at the end of this document.

Computations done in imaginary time can yield an excited state energy by a transformation of the energy decay curve.

If an accurate description of the ground state is already available, an excited state description can be obtained by forcing the wave function to be orthogonal to the ground state wave function.

Diffusion and Green's function QMC calculations are often done using a fixed node approximation. Within this scheme, the nodal surfaces used define the state that is obtained as well as ensuring an antisymmetric wave function.

Matrix QMC procedures similar to configuration interaction treatments have been devised in an attempt to calculate many states concurrently. These methods are not yet well developed as evidenced by oscillatory behavior in the excited state energies.

Path integral methods

Further information

For an introduction to quantum chemistry see
D. A. McQuarrie "Quantum Chemistry" University Science Books (1983)

A graduate level text on quantum chemistry is
I. N. Levine "Quantum Chemistry" Prentice Hall (1991)

For quantum Monte Carlo methods, order the following book using ISBN 981-02-0322-5 because the title is listed incorrectly in 'Books in Print'.
B. L. Hammond, W. A. Lester, Jr., P. J. Reynolds "Monte Carlo Methods in Ab Initio Quantum Chemistry" World Scientific (1994)

For density functional theory see
R. G. Parr, W. Yang "Density-Functional Theory of Atoms and Molecules" Oxford (1989)

There is a comprehensive listing of all available molecular modeling software and structural databanks, free or not, in appendix 2 of
"Reviews in Computational Chemistry Volume 6" Ed. K. B. Lipkowitz and D. B. Boyd, VCH (1995)

For an introduction to excited states via path integral methods see
T. E. Sorensen, W. B. England Molecular Physics, 89, 1577 (1996)

An expanded version of this article will be published in "Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems" by David Young, which will be available from John Wiley & Sons in the spring of 2001.

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