Calculating Electronic Excited States

E-mail: dyoung@asc.edu

Division of University Computing
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Auburn University
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Introduction

This is an introduction to the techniques used for the calculation of electronic excited states of molecules (sometimes called eximers). Specifically, these are methods for obtaining wave functions for the excited states of a molecule from which energies and other molecular properties (such as dipole moments) can be calculated. These calculations are an important tool for the analysis of spectroscopy, reaction mechanisms and other excited state phenomena.

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

Ab initio programs attempt to compute the lowest energy state of a specified multiplicity. Thus calculations for different spin states will give a lowest energy state and a few of the excited states. This is most often done to determine singlet - triplet gaps in organic molecules.

CIS

A single excitation configuration interaction (CIS) calculation is probably the most common way to get excited state energies. This is because it is one of the easiest calculations to perform.

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

If the initial guess for a calculation is very close to an excited state wave function, the calculation may converge to that excited state. This is typically done by doing an initial calculation then using it's wave function with some of the orbitals switched as the initial guess for another calculation.

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

Most ab initio calculations use symmetry adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal meaning that every molecular orbital will behave according to the symmetry of one of the irreducible representations of the point group. No orbitals will be described by a mixing of different irreducible representations.

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

For configuration interaction calculations of double excitations or higher, it is possible to solve the CI super-matrix for the second root, third root, fourth root, etc.

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

Some programs, such as COLUMBUS, allow a calculation to be done with some orbitals completely neglected from the calculation. For example, in a transition metal compound you could work with four d functions so that the calculation would have no way to occupy the function that was left out.

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

Traditionally, excited states have not been one of the strong points of density functional theory. This is due to the difficulty in ensuring orthogonality to the ground state wave function when no wave functions are being used in the calculation.

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

Quantum Monte Carlo (QMC) methods are computations which use a statistical integration to calculate integrals which could not be evaluated analytically. These calculations can be extremely accurate, but often at the expense of enormous CPU times.

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

There has been some initial success at computing excited state energies using the path integral formulation of quantum mechanics (Feynman's method). In this formulation the energies are computed using perturbation theory. There has not yet been enough work in this area to give any general understanding of the reliability of results or relative difficulty of performing the calculations. However, the work that has been done indicates that this may in time be a viable alternative to the other methods mentioned here.

Further information

There are many books on the principles of quantum mechanics and every physical chemistry text has an introductory treatment. The work which I am listing here is a two volume set with each chapter broken into a basic and advanced sections making it excellent for both intermediate and advanced users.
C. Cohen-Tannoudji, B. Diu, F. Laloe "Quantum Mechanics Volumes I & II" Wiley-Interscience (1977)

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)

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E-mail David Young at dyoung@asc.edu