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.  Approximate wave functions also have great difficulty describing the energies and properties of molecules in stretched or non-equilibrium geometries, and yet it is precisely these geometries that are critically important to understanding chemical reactivity. 

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, which John Coleman 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, known as positivity 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,1-5 and (ii) minimization of the ground-state energy as a functional of the 2-RDM only, known as the variational 2-RDM method.6,7 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. Within the 1-RDM the signature of multi-reference correlation is the appearance of eigenvalues (natural occupation numbers) that are far from zero (unoccupied) or one (occupied). The ACSE offers an ideal framework for treating multi-reference effects. 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. Unlike other multi-reference methods, the ACSE also yields 2-RDMs for a direct calculation of the properties and property surfaces.

Selected References

1. 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).
2. 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).
3. 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).
4. 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).
5. D. A. Mazziotti, “Determining the energy gap between the cis and trans isomers of HO₃^- using geometry optimization within the anti-Hermitian contracted Schrödinger and coupled cluster methods,” J. Phys. Chem. A 111, 12635 (2007).
6. D. A. Mazziotti, “Quantum chemistry without wavefunctions: Two-electron reduced density matrices,” Invited Article, Acc. Chem. Res. 39, 207 (2006).
7. 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).