Chemical Dynamics in the Gas Phase

C. Theoretical Studies of Potential Energy
Surfaces and Computational Methods

Ron Shepard

Research Objectives

This project involves the development, implementation, and
application of theoretical methods for the calculation and
characterization of potential energy surfaces (PES) involving
molecular species that occur in hydrocarbon combustion. These
potential energy surfaces require an accurate and balanced
treatment of reactants, intermediates, and products.

Approach

Most of our work focuses on general multiconfiguration
self-consistent-field (MCSCF) and multireference single-
and double-excitation configuration interaction (MRSDCI)
methods. In contrast to the more common single-reference
electronic structure methods, this approach is capable of
describing accurately molecular systems that are highly
distorted away from their equilibrium geometries, including
reactant, fragment, and transition-state geometries, and of
describing regions of the potential surface that are associated
with electronic wave functions of widely varying nature. The
MCSCF reference wave functions are designed to be
sufficiently flexible to describe qualitatively the changes in
the electronic structure over the broad range of molecular
geometries of interest. The necessary mixing of ionic, covalent,
and Rydberg contributions, along with the appropriate
treatment of the different electron-spin components (e.g.
closed shell, high-spin open-shell, low-spin open shell,
radical, diradical, etc.) of the wave functions are treated
correctly at this level. Further treatment of electron correlation
effects is included using large scale multireference CI wave
functions, particularly including the single and double
excitations relative to the MCSCF reference space. This
leads to the most flexible and accurate large-scale MRSDCI
wave functions that have been used to date in global PES
studies.

Highlights of Recent Work
(November 1998 - November 2001)

Electronic Structure Code Development and Maintenance.
A major component of this project is the development and
maintenance of the COLUMBUS Program System. The
COLUMBUS Program System computes MCSCF and
MRSDCI wave functions, MR-ACPF (averaged coupled-
pair functional) energies, MR-AQCC (averaged quadratic
coupled cluster) energies, spin-orbit CI energies, and analytic
energy gradients. Geometry optimizations to equilibrium and
saddle-point structures can be done automatically for both
ground and excited electronic states.

Maintenance and Distribution. The COLUMBUS Program
System is maintained and developed collaboratively with
several researchers including Isaiah Shavitt and Russell M.
Pitzer (Ohio State University), and Hans Lischka (University
of Vienna, Austria). The COLUMBUS Program System of
electronic structure codes has been maintained on the various
machines used for production calculations by the Argonne
Theoretical Chemistry Group, including IBM RS6000
workstations, DEC/COMPAC ALPHA workstations, the
parallel IBM SP QUAD machine at ANL, the parallel IBM
SP at NERSC, and the Theoretical Chemistry Group’s IBM
SP parallel supercomputer. Most recently, the codes have
been ported to the Argonne 512-CPU Linux cluster, Chiba
City, and the codes will be ported to the new Theoretical
Chemistry Group Linux cluster that will be installed in the
next few months. The COLUMBUS codes also have been
ported recently to the Macintosh personal computer, allowing
sophisticated production-level electronic structure calculations
on desktop and laptop computers. These computer codes are
used in production-level molecular applications by members
and visitors of the Argonne Theoretical Chemistry Group.

The COLUMBUS Program System is available using the
anonymous ftp facility of the internet. The codes and online
documentation are now also available from the web address
http://www.itc.univie.ac.at/~hans/Columbus/columbus.html. The
latest code version, 5.7, was released in April, 2001, and a
new version 5.8 is imminent. In addition to the source code,
the complete online documentation, installation scripts, sample
calculations, and numerous other utilities are included in the
distribution. A partial implementation of an IEEE POSIX
1009.3 library has been developed and is also available from
the anonymous ftp server ftp.tcg.anl.gov. This library
simplifies the porting effort required for the COLUMBUS
codes, and also may be used independently for other Fortran
programming applications. Major components of the
COLUMBUS Program System are also distributed as part
of the larger NWCHem electronic structure package.

Parallelization Developments. In collaboration with Hans
Lischka (University of Vienna, Austria) and Robert Harrison
(Pacific Northwest National Laboratory), the parallel version
of the CI diagonalization program CIUDG has been developed
and ported to several large parallel machines. Using the
TCGMSG and MPI libraries, this program also runs on
networks of workstations, small-scale shared-memory parallel
machines (e.g. Cray), and small-scale distributed memory
machines (e.g. Intel IPSC/i860 and Linux clusters). The latest
version of this parallel code uses the Global Array library, the
use of which eliminates unnecessary synchronization steps
from earlier versions of this code and reduces the overall
communications requirements for larger numbers of nodes.
Excellent scalability on as many as 320 nodes of the Intel
Delta, 256 nodes of the IBM SP, and 512 nodes on the
Cray T3D and T3E have been demonstrated, and calculations
on the 512-CPU Linux Cluster at Argonne are in progress.
One benchmark calculation by Dachsel, et al [J. Phys. Chem.
A, 103, 152-155 (1999)] using this code employed a 1.3E9
CSF expansion of the wave function, the largest MRSDCI
wave function ever reported. This is the first successful attempt
to parallelize a production-level MRSDCI code, and this effort
represents a major step forward toward using effectively the
large-scale parallel supercomputers that are becoming available
to scientists. Generalizations of the method are planned that will
allow treatment of larger molecular systems. Future effort will
be directed also to integrate the parallel version of the code
with other parts of the COLUMBUS Program System to allow
production-level PES calculations.

Initial applications of the MRSDCI analytic energy gradient code
have begun and have included both ground and excited electronic
states, and a variety of states including both closed- and open-shell
systems and states of mixed valence and Rydberg character. A
study of geometry optimizations for several small molecules has
been initiated with the current code. This allows direct comparisons
with experimental results and with other electronic structure
methods. These geometry comparisons demonstrate that even
with modest MCSCF reference spaces, the MRSDCI geometries
compare very well with the very best single-reference methods
currently in use. This is encouraging because the flexible MRSDCI
wave functions are expected to be comparably accurate over the
entire PES, whereas it is known that the accuracy of single-
reference methods is biased toward those regions of the PES
that are dominated by the reference determinant (such as near
equilibriumconformations). These initial comparisons are for
ground-state singlet molecules, and future work will include
similar comparisons for high-spin states, radicals, and excited
electronic states. Another recent application has been the study
of the 1¹B₁(s ^-p*) and the 2¹A₁(p ^-p*) excited states of
formaldehyde. It was found that, with flexible and accurate
multireference wave functions, these two states form a conical
intersection (see Fig. 1 shown as a .pdf file). Optimization of
the geometry using analytic energy gradient methods reveals
a nonlinear structure, in contrast to the findings of previous
theoretical studies of this state. The critical feature of the
electronic structure that must be correct in order to describe
this conical intersection accurately is the elimination of spurious
charge contamination in the C–O bond.

The major computational step involved in MRSDCI analytic
energy gradients after the energy calculation is the computation
of the CI reduced density matrices. This step typically requires
about 10% of the effort of the energy calculation. Very large-
scale wave functions can be optimized with the parallel
MRSDCI code, reducing the (wall clock) time required for
energy calculations considerably. Parallelization of the CI density
step is then required to maintain a 10% overhead cost for
gradients beyond the single point energy cost. In the past year,
a parallel version of the CI density code has been developed
that is based on the same basic methodology that is used in the
parallel CI code. The first applications have been initiated with
this code–the geometry optimization of C₂H₄ with a flexible
multireference wave function using three orbital basis sets. The
bond lengths are given in Table I for three basis sets for the
MCSCF wave function and for two basis sets (the largest basis
set calculation is in progress) for the MRSDCI wave function.
These bond lengths are extrapolated to the infinite basis set limit
in both cases and compared to the spectroscopic experimental
values. When completed, these will represent the largest-scale
MRSDCI calculations with analytic energy gradient optimizations
ever computed.

Table I. C₂H₄ Summary

†The 3-point MCSCF basis set extrapolation is based on the
exponential model: R_n=R_infinity+AB^-n. The 2-point MRSDCI
extrapolation is based on the geometric model: R_n=R_infinity+An^-3.

Iterative Matrix Diagonalization: A new iterative subspace
diagonalization method, called Subspace Projected Approximate
Matrix (SPAM), has been developed. In a subspace method, a new
trial vector is added to an existing vector subspace in each iteration.
The choice of expansion vectors determines the convergence rate.
The traditional Davidson and Lanczos methods are examples of
iterative subspace methods. In the SPAM approach, an approximate
matrix is constructed with each iteration using a projection operator
approach, and the eigenvector of this approximate matrix is used
to define the new expansion vector. The convergence rate is
improved over the Davidson and Lanczos approaches by
choosing an appropriate approximate matrix to define the
expansion space. The efficiency of the procedure depends on the
relative expense of forming approximate and exact matrix-vector
products. In the past year this method has been extended in two
different ways. First, it has been extended to simultaneous
optimization of several roots. This is achieved by converging all
of the roots at the approximate level before contracting and
computing the exact matrix-vector products; this minimizes the
overall effort required to optimize all of the desired eigenpairs.
Second, the method has been extended to allow an arbitrary
number of levels of matrix approximations. This results in a
multiroot-multilevel SPAM algorithm that has a wide range
of possible applications. This matrix diagonalization method
has been applied to a wide range of eigenvalue problems,
including optimization of the lowest eigenpairs, the highest
eigenpairs, and interior eigenpairs using both root-homing and
vector-following. Approximations have been generated by
neglecting small matrix elements, by tensor-product approximation,
by expansion truncation, by neglect of off-diagonal matrix blocks,
and by operator approximation. These applications demonstrate
the wide range of applicability of the SPAM diagonalization
method. This code is available from the anonymous ftp server
ftp.tcg.anl.gov. Work in progress is directed toward
approximation of MRSDCI Hamiltonian matrices based on
repulsion integral approximation. We have been invited to publish
this work in a Feature Article in Computer Physics Communications.

Publications, Submissions, and Talks (1998 – 2001)

Publications and Submissions

"A Systematic Ab Initio Investigation on the Open and Cyclic
Structures of Ozone," T. Müller, S. S. Xantheas, H. Dachsel,
R. J. Harrison, J. Nieplocha, R. Shepard, G. S. Kedziora, and
H. Lischka, Chem. Phys. Letters 293, 72-80 (1998)

"High-Performance Computational Chemistry: Hartree-Fock
Electronic Structure Calculations on Massively Parallel
Processors," J. L. Tilson, M. Minkoff, A. F. Wagner, R. Shepard,
P. Sutton, R. J. Harrison, R. A. Kendall, A. T. Wong, The
Int. J. High Performance Computing Applications 13,
291-302 (1999)

"Ab Initio Determination of Americium Ionization Potentials,"
J. L. Tilson, R. Shepard, C. Naleway, A. F. Wagner, and
W. C. Ermler, J. Chem. Phys. 112, 2292-2300 (2000)

"High-Level Multireference Methods in the Quantum-Chemistry
Program System COLUMBUS: Analytic MR-CISD and
MR-AQCC Gradients and MR-AQCC-LRT for Excited States,
GUGA Spin-Orbit CI, and Parallel CI," H. Lischka, R. Shepard,
R. M. Pitzer, I. Shavitt, M. Dallos, T. Muller, P. G. Szalay,
M. Seth, G. S. Kedziora, S. Yabushita, and Z. Zhang, Phys.
Chem. Chem. Phys. 3, 664-673 (2001)

"Geometry Optimization of Excited Valence States of
Formaldehyde Using Analytical Multireference Configuration
Interaction Singles and Doubles and Multireference Averaged
Quadratic Coupled-Cluster Gradients, and the Conical Intersection
Formed by the1¹B₁(s-p^*) and the 2¹A₁(p-p^*) States,"
M. Dallos, T. Muller, H. Lischka, and R. Shepard, J. Chem. Phys.
114, 746-757 (2001)

"The Calculation of f-f Spectra of Lanthanide and Actinide Ions
by the MCDF-CI Method," M. Seth, K. G. Dyall, R. Shepard,
and A. Wagner, J. Phys. B: At. Mol. Opt. Phys. 34, 2383-2406
(2001)

"The Subspace Projected Approximate Matrix (SPAM)
Modification of the Davidson Method," R. Shepard, A. F. Wagner,
J. L. Tilson, and M. Minkoff, J. Comp. Phys. 172, xxx
(in press)

"An Ab Initio Study of the Ionization Potentials and the f-f
Spectroscopy of Europium Atoms and Ions," C. Naleway,
M. Seth, R. Shepard, A. F. Wagner, J. L. Tilson, W. C. Ermler,
and S. R. Brozell (in press)

"Analytic MRCI Gradient for Excited States: Formalism and
Application to the n-p* Valence- and n-(3s,3p) Rydberg
States of Formaldehyde,” H. Lischka, M. Dallos, and R. Shepard,
Mol. Phys. (in press)

"Reducing I/O Costs for the Eigenvalue Procedure in
Large-Scale CI Calculations," R. Shepard, I. Shavitt, and
H. Lischka, J. Computational Chem.. (in press)

"An Ab Initio Study of the f-f Spectroscopy of Americium⁺³,"
J. L. Tilson, C. Naleway, M. Seth, R. Shepard, A. F. Wagner,
and W. C. Ermler, J. Chem. Phys. (in press)

"A Study of Molecular Bondlengths Using Multireference
Configuration Interaction Methods," G. S. Kedziora, R. Shepard,
M. Seth, H. Lischka, T. Mueller, and M. Dallos (in preparation)

Talks and Presentations

Symposium on Combustion Chemistry/20th Combustion
Research Conference, Invited Participant, Contributed Poster,
"Analytical Energy Gradients for CI Wave Functions," 215
American Chemical Society National Meeting, Physical
Chemistry Symposium, Dallas, Texas (1998)

Electronic Structure Theory: From Methods to Molecules and
Materials, Invited Participant, "Analytical Energy Gradients for
CI Wave Functions,"American Physical Society Centennial
Meeting, Division of Chemical Physics Symposium, Atlanta,
Georgia (1999)

Ohio State University International Symposium on Molecular
Spectroscopy 55th Meeting, Invited Participant and Session
Chair, Columbus, Ohio (1999)

National Science Foundation Information Technology Research
Review Panel, Invited Participant, Washington, D.C. (2000)

21^st Combustion Research Conference, Participant, Chantilly,
Virginia, (2000)

University of Buffalo, Center for Computational Research,
Invited seminar, "The Subspace Projected Approximate Matrix
(SPAM) Modification of the Davidson Method," Buffalo,
New York (2000)

Sanibel Symposium 2001, Participant, "The Subspace Projected
Approximate Matrix (SPAM) Modification of the Davidson
Diagonalization Method," St. Augustine, Florida (2001)

22^nd Combustion Research Conference, Invited Participant,
"Theoretical Studies of Potential Energy Surfaces and
Computational Methods," Tahoe City, California (2001)

Molecular Quantum Mechanics: The Right Answer for the
Right Reason Participant, "The Subspace Projected Approximate
Matrix (SPAM) Modification of the Davidson Diagonalization
Method," and "Analytic MR-CISD and MR-AQCC Gradients:
Full Optimization of Minima and Saddle Points on Potential
Energy Surfaces for Valence-Excited States of Formaldehyde
and Acetylene," Seattle, Washington (2001)

RON SHEPARD

Office Address:
Chemistry Division
Argonne National Laboratory
9700 South Cass Avenue
Argonne, IL 60439
630-252-3584 Fax: 630-252-4470
E-mail: shepard@tcg.anl.gov

Education:

University of Utah, Salt Lake City, Ph.D. in Physical
Chemistry, 1980

Calhoun-Halkjaer Award for Outstanding Research (1979) (Research Advisor: Professor J. Simons)

University of Central Arkansas, B.S. with Majors in
Mathematics and Chemistry, 1975, Trinity Foundation
Scholar (1970-1975) (Research Advisor: Professor
J. M. Manion)

Battelle Columbus Laboratories, Columbus, Ohio,
Post Doctoral, 1980

Battelle Technical Development Post Doctoral Fellowship
(Research Advisor: Professor I. Shavitt)

Positions and Research Experience:

July 1986 - Present Scientist, Argonne National Laboratory

November 1981 - July 1986 Assistant Scientist, Argonne
National Laboratory

January 1981-October 1981 Research Scientist, Battelle
Columbus Laboratories, Columbus, Ohio

Professional Organizations:

Member of the American Chemical Society, Division
of Physical Chemistry

Member of the Argonne Computer Users Group

Member of the Argonne Macintosh Users Group
(Vice President 1987 – Present)

Member of Sigma Xi, The Scientific Research Society

Member of the Gaussian Scientific Advisory Board
(1993 – 1997)

Member of the Chemistry Division Computer Planning
Committee (1996-Present)

Return to Chemical Dynamics in the Gas Phase

Glassblowing

Interfacial Processes

Radiation and
Photochemistry

Photosynthesis
Biological Materials Growth Facility

Cluster Studies

Chemical Dynamics

Atomic Physics

Nanophotonics

Heavy Elements

Coordination Chemistry

f-Electron Interactions

Actinide Facility

Computational Materials and Electrochemical Processes