The Approximate Inclusion of Triple Excitations in EOM-type Quantum Chemical Methods

Michael Rust
Advisor: Robert Cave

The non-relativistic theory of quantum mechanics provides a description of atomic physics that, in principle, should be able to predict all the empirically observed features of chemistry.  Mathematically, this description is stated as an eigenvalue problem for an operator H, closely related to the classical Hamiltonian for the system under consideration.  The eigenvalues are the allowed energies of the system, and the eigenvectors are elements of an infinite-dimensional Hilbert space describing the state in question.  Perfect knowledge of these eigenvectors would permit the calculation of any desired physical properties (bond lengths, dipole moments, etc...)

This program works spectacularly for the hydrogen atom, a system for which the Hamiltonian is simple enough that the eigenvalue problem results in a differential equation that can be solved by separation of variables.  Unfortunately, no one has been able to figure out how to solve the more complicated eigenvalue problems resulting from molecules with more than a single electron.

Enter approximate techniques.  The simplest tactic is a kind of average field approximation originally due to Hartree and Fock.  In the Hartree-Fock scheme we iteratively obtain approximate eigenvectors that are the "best" possible wavefunctions formed by products of single-electron wavefunctions.  Here "best" is meant in a variational sense, since it is easily shown that the true ground state has the property that it has a lower expectation for the energy than any other conceivable state.  In essence, we begin with a guess for the electron density and then converge interatively to a self-consistent, variationally optimal set of single-electron wavefunctions.  For an n electron molecule, the Hartree-Fock ground state is formed by taking a product of the lowest n single-electron wavefunctions.

Depending on your perspective, Hartree-Fock is either a stunning success or miserably inadequate.  While a Hartree-Fock solution gets the energy "almost right" in an absolute sense, many important problems in chemistry are sensitive to the subtle differences in energy between closely related states.  As a practical example, consider the fluorine molecule.  Though the Hartree-Fock energy is very close to the true energy, geometry calculations based on Hartree-Fock show, disturbingly, that molecular fluorine should not exist at all (i.e. the lowest energy can be obtained by infinitely separating the fluorine atoms).

The problem, of course, is that the true eigenvectors are not products of single-electron wavefunctions.  Classically, we can interpret the failings of Hartree-Fock as a lack of proper correlation between electron trajectories: when the electron interaction is averaged over there is nothing that guarantees that individual electrons will be well-separated spatially.  For this reason, more advanced methods that build upon a Hartree-Fock reference are known as "correlated methods."

The conceptually simplest correlated method is known (opaquely) as "configuration interaction" or CI.  When we formed the Hartree-Fock ground state, we filled the lowest n single-electron orbitals.  In principle, however, the Hartree-Fock average field operator has an infinite set of eigenvectors, the n-electron products of which must span the entire Hilbert space containing the true ground state.  In a CI calculation, we approximate the true ground state by combining "excited" states formed by reomving electrons from the lowest n wavefunctions and populating the higher unoccupied orbitals.  Unfortunately, CI suffers from two severe problems.  First, it is computationally intractable to include many configurations in the calculation.  For many practical molecules, only single excitations can be plausibly included.  Secondly, CI suffers from size-consistency issues that cripple its general usefulness.

Finally, we come to the title of my proposal and original work.  The (even more obliquely named) Equation-of-Motion (EOM) method circumvents some of these limitations of CI in a clever manner.  By applying a non-linear transformation to the Hamiltonian, it is possible to generate excitations in an "exponential" manner, simultaneously addressing the issue of size-inconsistency and permitting an inclusion of higher-order configurational effects through the calculation of only lower order excitations.

EOM is arguably the most powerful and generally successful method available today for quantum chemical calculations.  For reasons of computational complexity, all current implementations include only single and double excitations.  Though this seems to be adequate for most problems, there are important cases where higher-order excitations are important.  I want to attempt to develop an extension of EOM which includes the most important triple excitations in an approximate way.  Ideally, this can make a significant impact on some problems with a marginal increase in computational cost.  I plan to both investigate the theoretical properties of such an extension to the EOM method and to implement it using the SGI's in Dr. Cave's lab.

I have spent the fall semester of 1999 and the summer of 2000 working with Prof. Cave and doing research related to these and other quantum chemistry issues. I have written a Hartree-Fock implementation and have studied the source code to the existing implementation of the EOM method. Next semester I will be enrolled in Prof. Gu's linear algebra class (Math 173) which should prove useful in the theoretical analysis of this problem.

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| Math Department | Last modified: July 2000