Parallel Calculation of Electron-Transfer and Resonance Matrix

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on April 10, 2018 | https://pubs.acs.org Publication Date: May 17, 1995 | doi: 10.1021/bk-1995-0592.ch007

Chapter 7

Parallel Calculation of Electron-Transfer and Resonance Matrix Elements of Hartree—Fock and Generalized Valence Bond Wave Functions Erik P. Bierwagen, Terry R. Coley, and William A. Goddard, III Materials and Molecular Simulation Center, Beckman Institute, California Institute of Technology, 139-74, Pasadena, CA 91125

We review the theory for the computation of the Hamiltonian matrix element between two distinct electronic wave functions Ψ and Ψ sharing the same nuclear configuration but differing electronic density distributions. For example, Ψ and Ψ might describe two endpoints in an electron transfer reaction or two configurations in a reso nance description of a molecule. In such cases the calculation of the rate of electron transfer or resonance energy requires evaluation of A

Α

B

B

ΨA|Ĥ|ΨB = Ψ matrix elements. Because the orbitals of Ψ and Ψ have complicated (non-orthogonal) relationships, the calculation of H had been computationally intensive. In this paper we consider Ψ , Ψ having the form of closed or open-shell Hartree-Fock or Generalized Valence Bond wave functions and show the parallel structure of the theory. Using this parallel structure we present an ef ficient computational implementation for shared memory multi processors. AB

Α

B

AB

Α

B

The starting point of most ab initio quantum chemistry is an antisymmetrized product of molecular orbitals Ψ = |0 0$0 ...|. To compute properties such as energy, Α

A

C

E = (Ψ \Ê\ Ψ ), the molecular orbitals of Ψ are constructed to be mutually or thogonal. However, many problems are conveniently described in terms of two dif ferent ground state wave functions. For example to describe the charge transfer be tween Ψ and Ψβ\ A

Α

Α

Α

Α

0097-6156/95/0592-0084$12.00/0 © 1995 American Chemical Society

Mattson; Parallel Computing in Computational Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

7. BIERWAGEN ET AL.

85

Hartree-Fock and GVB Wave Functions

we need to compute cross-matrix elements, ( ¥ Ά | ^ | ^B) lecular orbital of Ψ overlaps some or all orbitals of Ψ . transfer rate is proportional to I ^ A ^ > where: Α

=

^ A B » where each mo In this case the electron

Β

2

Τ

-

H

AB

S

~ 1

H

AB AA

(1)

S


~ AB

=

Α

8

i s t h e

SAB ( *Ρ I Ψ ) overlap matrix. Another example is the computation of chemical resonance energies. In this case Ψ and Ψ describe two different valence states (e.g., the two valence states of benzene). Representing the resonating wave function as Ψ = Ψ + Ψ , we can cal culate its energy, Α

B

Α

Α

Β

_(Ψ + Ψ \Η\Ψ* + Ψ*)^Η E^AB= ^Ψ + Ψ j Ψ + Ψ^

+Η

ΑΑ

Am

Α

Β

Α

+ 2Η

ΒΒ

Α

2 + 2S

Β

> ^ >

Β

3

a

A

b

(

2

)

only i f we have the means to calculate H and S ^ . Because of its intimate rela tionship to resonance energies, we will refer to H as a resonance matrix element. The computation of resonance matrix elements can also be used to evaluate con figuration interaction (CI) wave functions in cases where the configurations are nonorthogonal. Such non-orthogonal CIs have been successfully carried out (7); how ever, the computational complexity has limited the applications. If they can be made practical, non-orthogonal CI approaches have two distinct advantages over orthogo nal CIs: 1) the component states Ψ and Ψ can be chosen to be chemically meaningful descriptions of the system 2) this "better" choice of basis states reduces the number of states needed to accurately describe the system Electronic reorganization problems such as electron transfer and interpretation of photoelectric spectra lead naturally to a few-state description in terms of non-or thogonal basis states. A straightforward calculation of for non-orthogonal wave functions in volves non-orthogonal matrix elements involving all orbitals of Ψ overlapping all orbitals of Ψ leading to an Ν ! dependency, where Ν is the number of occupied spatial orbitals in each wave function. This contrasts with the case of orthogonal spatial orbitals where there are only of order Ν operations. To simplify this problem Voter and Goddard (2) showed that a pair of unitary transformations exists, which when applied to the molecular orbitals of Ψ and Ψ , respectively, a) leave the to tal energy, E^, unchanged and b) reduce the computational effort to order Ν by transforming Ψ and Ψ such that each orbital of Ψ overlaps exactly one orbital of Ψ . By reducing the computational effort this biorthogonalization, makes the resonance calculation tractable. Despite the computational savings obtained with clever transformations such as biorthogonalization, many systems of interest, especially in electron transfer studies, remain too large for practical H calculations with existing computer codes. Cur rent programs, which have served well for smaller cases, do not exploit the underlyAB

AB

Α

Β

Α

Β

2

Α

Β

2

Α

Β

Α

Β

AB


86

PARALLEL COMPUTING IN COMPUTATIONAL CHEMISTRY


ing parallelism in the theory and therefore cannot take advantage of multi-processor computers without significant restructuring (3). Goals Through the use of modern programming languages and software design we have produced a program for computing resonance matrix elements for systems of potentially unlimited size. The program meets the following design goals: • efficient performance on shared-memory multi-processors • user level control over the program's internal data structures and algorithms The Method section exposes the parallelism inherent in the theory of resonance ma trix element calculations. The Algorithm section introduces algorithms for sharedmemory multi-processors and shows how the first goal was achieved. Program Ar chitecture discusses our second goal in more general terms. Finally, in Results and Discussion we present timings and resonance energies for two systems of chemical interest. Method The computational theory of resonance matrix elements was developed by Voter and Goddard to examine the resonance energy between valence bond (and generalized valence bond (GVB)) (4-7) wave functions and is described elsewhere (2, 8). The following discussion highlights those parts of the theory that assist in understanding the parallel algorithm. Consider the resonance energy between two HF type wave functions as in equation 2 χ

where Ψ = \φΪΦ*ΦΪ··\

is a normalized, antisymmetrized molecular wave function,

and 0.*are the molecular orbitals (MO's). This problem is simplified by transform A

Β

ing the orbitals of *F and Ψ such that: (3) This biorthogonalization reduces the problem to the more standard-looking evalua tion: Α

Β

Η =(ψ \Η\ψ )3ηάΞ ^Υ[λ ΑΒ

ΑΒ

{

(4) where the overlap has been replaced by a product of the individual orbital overlaps. Expanding the above expression over molecular orbitals leads to:

(5) (6) (7) ^Γ=(^(1)^(2)|-|^(1)^(2)) r > the exchange term λΊ


(8)

7. BIERWAGEN ET AL.

87

Hartree-Fock and GVB Wave Functions

Generalizing this result to the open-shell case requires treating the alpha and beta spin systems separately when performing the biorthogonalization. This treatment is necessary in order to produce transformations which leave the energy unchanged: H

K

AB ~ Σ Άΐα^ία,ία + Σ Ήΐβ^ί^ΐβ + Σ Viaja[jîa,ja ia ίβ iaja Σ

Β


Φ>]β

Κ

νίβ^φ ;β- φ*β)+

Σ

" ia,ja ) + B

iajfi

ViaJfi{j?a jv)

(9)

where the a and β indices indicate spin, and the r/'s are as defined in equation 5. Further generalizing this result for multi-determinantal wave functions: φ A _ ^ çAa ψΑα

φΒ

^ ç

Parallel Calculation of Electron-Transfer and Resonance Matrix

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