Molecular second-quantized Hamiltonian: Electron correlation and non-adiabatic coupling treated on an equal footing

Abstract

We introduce a new theoretical and computational framework for treating molecular quantum mechanics without the Born-Oppenheimer approximation. The molecular wavefunction is represented in a tensor-product space of electronic and vibrational basis functions, with electronic basis chosen to reproduce the mean-field electronic structure at all geometries. We show how to transform the Hamiltonian to a fully second quantized form with creation/annihilation operators for electronic and vibrational quantum particles; paving the way for polynomial-scaling approximations to the tensor-product space formalism. In addition, we make a proof-of-principle application of the new ansatz to the vibronic spectrum of C 2 .

Quantum Proton Effects from Density Matrix Renormalization Group Calculations

“…In order to find an expression for the Fock operators, we define Coulomb operator, J is,µ (1), and exchange operator, K is,µ (1), in analogy to the Hartree-Fock method of electronic structure theory. With the two-body interaction, g, the Coulomb operator is written as…”

“…It is convenient to calculate the expectation value, Ψ| O (n) |Ψ , of a transition operator starting from its second-quantized form. In the case of spin- 1 2 fermions, the transition operators in second quantization are presented in Ref. 76.…”

“…Whereas methods of many-particle quantum mechanics relying on the omnipresent Born-Oppenheimer (BO) approximation are cornerstones of quantum chemistry and molecular physics, the recent advances of methods that do not rely on the BO approximation highlight the relevance of nuclear quantum and non-adiabatic effects in many chemical phenomena, including hydrogen tunneling, vibrationally excited states, and proton transfer reactions. [1][2][3][4][5][6][7][8][9][10] In nuclear-electronic methods, nuclei are treated on an equal footing with electrons. As a consequence, approximations of eigenfunctions of the nuclear-electronic Hamiltonian include nonadiabatic couplings between nearly degenerate electronic states systematically, 11-14 and the zero-point vibrational energy level is obtained in a single calculation.…”

Constrained-Path Auxiliary-Field Quantum Monte Carlo for Coupled Electrons and Phonons

“…[37][38][39] (It is also different from the coupled-cluster operators considered in recent studies of el-ph problems . [47][48][49] ) Given the performance improvement with the LLF trial wave function (as discussed below), we expect an el-ph Jastrow trial wave function will greatly reduce the difficulties in parameter regimes with strong el-ph coupling, and result in a major improvement in our AFQMC approach. We leave the implementation and systematic studies using an el-ph Jastrow trial wavefunction in AFQMC for future work.…”

“…Several methods have been formulated or extended to coupled el-ph problems, including DMRG, [30][31][32][33][34][35] variational exact diagonalization, 36 variational Monte Carlo, [37][38][39] dynamical mean-field theory, [40][41][42][43][44] density matrix embedding theory, 45,46 and coupled-cluster theory. [47][48][49] There are difficulties facing each approach. For example, large el-ph couplings and/or small phonon frequencies are challenging to handle in most methods based on a second quantized representation of phonons, because of the necessity of truncating the phonon Hilbert space.…”