Sum-of-products form of the molecular electronic Hamiltonian and its application within the MCTDH method

A first principles quantum formalism to describe the non-adiabatic dynamics of electrons and nuclei based on a second quantization representation (SQR) of the electronic motion combined with the usual representation of the nuclear coordinates is introduced. This procedure circumvents the introduction of potential energy surfaces and non-adiabatic couplings, providing an alternative to the Born–Oppenheimer approximation. However, the major problem of this method applied to ab initio studies of large molecular systems remains the enormous size of the electronic SQR Hamiltonian, whose number of terms increases with the fourth power of the number of spin-orbitals. We introduce three different approaches to represent the second-quantized electronic Hamiltonian in a sum-of-products form. These procedures aim at mitigating the quartic scaling of the number of terms in the Hamiltonian with respect to the number of spin orbitals, and thus enable applications to larger molecular systems. Here we describe the application of these approaches within the multi-configuration time-dependent Hartree framework.