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44695-AC6
Variational Reduced-Density-Matrix Calculations for Studying the Chemistry of Radicals
David A. Mazziotti, University of Chicago
My
research in reduced-density-matrix theory was initially inspired by Charles
Coulson's 50-year-old dream of using just two-electron reduced density matrices
instead of N-electron wave functions to describe the energies and properties of
many-electron atoms and molecules.
Coulson's challenge to chemists and applied mathematicians stemmed from
the awareness that computational work to compute the “exact” wave function of
an atom or molecule increases exponentially with the number of electrons. Wave function approximations with increasing
molecular size often fail to capture complex details of the exact wave function
that are needed to treat important chemical phenomena from van der Waals
interactions, energy splittings between spin states, and covalent bond
breaking.
Coulson's
dream was based on the fact that because the fundamental interactions between
electrons are two-body, the electronic energy and properties of any atom or
molecule may be readily expressed as a linear functional of only two-electron
variables, known as the two-electron reduced density matrix (2-RDM). However, developing a method based on the
2-RDM requires that certain rules be imposed upon the matrix to ensure that it
represents a realistic N-electron system.
The work of mathematicians and chemists helped to establish some rules
for constraining the 2-RDM, called N-representability conditions. Although small test calculations were
performed in the early years, they were unsuccessful because the full hierarchy
of N-representability conditions had not been derived and sufficient
computational resources in hardware and software did not exist.
Through
research in my group two complementary approaches to the direct calculation of
the 2-RDM without the wave function have emerged: (i) solution of the
anti-Hermitian contracted Schrödinger equation,1-4 and (ii) minimization of the
ground-state energy as a functional of the 2-RDM only, known as the variational
2-RDM method.5-7 These two methods are discussed in detail in a new book Reduced-Density-Matrix
Mechanics with Application to Many-electron Atoms and Molecules,1 which I
edited for the Advances in Chemical Physics series, published by John Wiley and
Sons.
In October of 2006 I announced in
Physical Review Letters a new approach to solving the contracted Schrödinger
equation that improves the accuracy in both the energy and the 2-RDM by an
order of magnitude. The contracted Schrödinger equation (CSE) is a projection
(or contraction) of the N-electron Schrödinger equation onto the space of two
electrons. The new approach, by solving only the anti-Hermitian part of the
contracted Schrödinger equation (ACSE), removes the previous limitations of the
CSE to yield between 95-100% of molecular correlation energy as well as highly
accurate one- and two-electron properties. The energies and properties from the
ACSE are more accurate than those from wave function methods of comparable
computational cost.
In recently published work the ACSE
method has been generalized to treat molecular systems with significant
multi-reference correlation in the wave function. The term multi-reference
indicates that the many-electron wave function has significant contributions
from more than one reference determinant. Such effects arise in bond breaking,
transition states, diradicals and biradicals, and transition-metal complexes.
While the scaling of the ACSE is similar to multi-reference perturbation
methods, it produces energies that are significantly more accurate as well as
potential energy surfaces with a lower non-parallelity error, and unlike other
multi-reference methods, the ACSE also yields 2-RDMs for a direct calculation
of the properties and property surfaces.
Under
the ACS-PRF Greg Gidofalvi and I have also developed an active-space version of
the variational 2-RDM method7 which is orders of magnitude more efficient in
both memory and floating-point operations than existing active-space wave
function methods. The method was
applied to acene chains, where chains longer than 4 cannot be treated by traditional
wave function methods.7 Results show
that as the chain length grows, the acene becomes a poly-radical.
Selected References
- Two-electron Reduced-Density-Matrix Theory for Many-electron Atoms and Molecules in Advances in Chemical Physics Series, D. A. Mazziotti, Editor (New York, Wiley, 2007).
- D. A. Mazziotti, “Anti-Hermitian contracted Schrödinger equation: direct determination of the two-electron reduced density matrices of many-electron molecules,” Phys. Rev. Lett. 97, 143002 (2006).
- D. A. Mazziotti,“Anti-Hermitian part of the contracted Schrödinger equation for the direct calculation of two-electron reduced density matrices,” Phys. Rev. A 75, 022505 (2007).
- D. A. Mazziotti,“Two-electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation: Enhanced energies and properties with larger basis sets,” J. Chem. Phys. 126, 184101 (2007).
- D. A. Mazziotti, “Quantum chemistry without wavefunctions: Two-electron reduced density matrices,” Invited Article, Acc. Chem. Res. 39, 207 (2006).
- G. Gidofalvi and D. A. Mazziotti, “Computation of dipole, quadrupole, and octupole surfaces from the variational two-electron reduced density matrix method,” J. Chem. Phys. 125, 144102 (2006).
- G. Gidofalvi and D. A. Mazziotti, “Active-Space Two-Electron Reduced-Density-Matrix Method: Complete Active-Space Calculations without Diagonalization of the N-Electron Hamiltonian,” J. Chem. Phys. (in press).
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