# Quantum Statistical Mechanics of Electronically Open Molecules: Reduced Density Operators

## TL;DR

This paper introduces a new way to describe the quantum behavior of molecules interacting with their environment by accounting for electron sharing.

## Contribution

A novel reduced density operator is proposed that includes particle-breaking interactions and resolves ambiguities in fermionic partial trace operations.

## Key findings

- A new reduced density operator is defined for electronically open molecules.
- The framework allows explicit electron occupancy in the environment at any theoretical level.
- The approach is compatible with various environment models and includes a generalized chemical potential.

## Abstract

We present a reduced density operator for electronically
open molecules
by explicitly averaging over the environmental degrees of freedom
of the composite Hamiltonian. Specifically, we include the particle-number
nonconserving (particle-breaking) interactions responsible for the
sharing of electrons between the molecule and the environment, which
are neglected in standard formulations of quantum statistical mechanics.
We propose an unambiguous definition of the partial trace operation
in the composite fermionic Fock space based on composite states in
a second quantization framework built from a common orthonormal set
of orbitals. Thereby, we resolve the fermionic partial trace ambiguity.
The common orbital basis is constructed by spatial localization of
the full orbital space, in which the full composite Hamiltonian naturally
partitions into a molecule Hamiltonian, an environment Hamiltonian,
and an interaction Hamiltonian. The new reduced density operator is
based on the approximation of commutativity between the subsystem
Hamiltonians (i.e., molecule and environment Hamiltonians) and the
interaction Hamiltonian, but our methodology provides a hierarchical
approach for improving this approximation. The reduced density operator
can be viewed as a generalization of the grand canonical density operator.
We are prompted to define the generalized chemical potential, which
aligns with the standard interpretation of the chemical potential,
apart from the possibility of fractional rather than strictly integer
electron transfer in our framework. In contrast to standard approaches,
our framework enables an explicit consideration of the electron occupancy
in the environment at any level of theory, irrespective of the model
used to describe the molecule. Specifically, our reduced density operator
is fully compatible with all possible level-of-theory treatments of
the environment. The approximations that render our reduced density
operator identical to the grand canonical density operator are (i)
restriction of excitations to occur within the same orbitals and (ii)
assumption of equal interaction with the environment for all molecule
spin orbitals (i.e., the wide-band approximation).