^{1}and David A. Mazziotti

^{1,a)}

### Abstract

Different sets of molecular orbitals and the rotations connecting them are of great significance in molecular electronic structure. Most electron correlation methods depend on a reference wave function that separates the orbitals into occupied and unoccupied spaces. Energies and properties from these methods depend upon rotations between the spaces. Some electronic structure methods, such as modified coupled electron pair approximations and the recently developed parametric two-electron reduced density matrix (2-RDM) methods [D. A. Mazziotti, Phys. Rev. Lett.101, 253002 (Year: 2008)]10.1103/PhysRevLett.101.253002, also depend upon rotations between occupied orbitals and rotations between unoccupied orbitals. In this paper, we explore the sensitivity of the ground-state energies from the parametric 2-RDM method to rotations within the occupied space and within the unoccupied space. We discuss the theoretical origin of the rotational dependence and provide computational examples at both equilibrium and non-equilibrium geometries. We also study the effect of these rotations on the size extensivity of the parametric 2-RDM method. Computations show that the orbital rotations have a small effect upon the parametric 2-RDM energies in comparison to the energy differences observed between methodologies such as coupled cluster and parametric 2-RDM. Furthermore, while the 2-RDM method is rigorously size extensive in a local molecular orbital basis set, calculations reveal negligible deviations in nonlocal molecular orbital basis sets such as those from canonical Hartree-Fock calculations.

D.A.M. gratefully acknowledges the National Science Foundation (NSF) CHE-1152425, the (U.S.) Army Research Office (USARO) (Grant No. W91 INF-1 1-504 1-346 0085), and Microsoft Corporation for their generous support.

I. INTRODUCTION

II. THEORY

A. p2-RDM method

B. Orbital invariance

III. RESULTS

A. Molecules at equilibrium

B. HF bond stretch

C. Non-interacting He atoms

IV. CONCLUSION

### Key Topics

- Wave functions
- 10.0
- Coupled cluster
- 7.0
- Ground states
- 7.0
- Mean field theory
- 6.0
- Electronic structure
- 5.0

## Figures

Potential energy surfaces for HF dissociation. The cc-pVQZ basis set was used.

Potential energy surfaces for HF dissociation. The cc-pVQZ basis set was used.

Energy per helium atom as a function of the number of helium atoms. The canonical orbitals used were found to be localized in both the occupied and virtual space. The cc-pVDZ basis set was used.

Energy per helium atom as a function of the number of helium atoms. The canonical orbitals used were found to be localized in both the occupied and virtual space. The cc-pVDZ basis set was used.

## Tables

The definitions for the topological factors are given by where n o is the number of occupied spin orbitals shared between {i, j} and {k, l} and n v is the number of virtual spin orbitals shared between {a, b} and {c, d}. The 9 possible combinations of n o /n v and the values of the topological factor for these combinations are listed in the table for the following three methods: configuration interaction doubles (CID), coupled electron pair approximation (CEPA(0)), and p2-RDM with the M parameterization. 17,18

The definitions for the topological factors are given by where n o is the number of occupied spin orbitals shared between {i, j} and {k, l} and n v is the number of virtual spin orbitals shared between {a, b} and {c, d}. The 9 possible combinations of n o /n v and the values of the topological factor for these combinations are listed in the table for the following three methods: configuration interaction doubles (CID), coupled electron pair approximation (CEPA(0)), and p2-RDM with the M parameterization. 17,18

The singlet ground-state energies of several molecules at equilibrium geometries. Relative energies for each method (or orbital rotation) are expressed as E method − E p2–RDM where E p2-RDM is the energy from the p2-RDM method with canonical Hartree-Fock orbitals. All calculations are performed in the cc-pVQZ basis set. Absolute energies are reported in Hartrees (H) and relative energies in milliHartrees (mH).

The singlet ground-state energies of several molecules at equilibrium geometries. Relative energies for each method (or orbital rotation) are expressed as E method − E p2–RDM where E p2-RDM is the energy from the p2-RDM method with canonical Hartree-Fock orbitals. All calculations are performed in the cc-pVQZ basis set. Absolute energies are reported in Hartrees (H) and relative energies in milliHartrees (mH).

The effect of orbital rotations on the p2-RDM method in the dissociation of HF. Relative energies (in mH) for each method are calculated according to E method − E p2-RDM. The cc-pVQZ basis set was used.

The effect of orbital rotations on the p2-RDM method in the dissociation of HF. Relative energies (in mH) for each method are calculated according to E method − E p2-RDM. The cc-pVQZ basis set was used.

Energy per helium atom in systems of non-interacting He atoms. He atoms are placed 200 Å apart. The cc-pVDZ basis set is used and energies are given in Hartrees.

Energy per helium atom in systems of non-interacting He atoms. He atoms are placed 200 Å apart. The cc-pVDZ basis set is used and energies are given in Hartrees.

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