Partially Wigner transformed Hamiltonian

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

Performing a partial Wigner transformation with respect to the nuclear variables, the molecular Hamiltonian can be written as... 

By performing a partial Wigner transform with respect to the coordinates of the environment, we obtain a classical-like phase space representation of those degrees of freedom. The subsystem coordinate operators are left untransformed, thus, retaining the operator character of the density matrix and Hamiltonian in the subsystem Hilbert space [4]. In order to take the partial Wigner transform of Eq. (1) explicitly, we express the Liouville-von Neumann equation in the Q representation,... 

The operator 4 = Vp -Vr —Vr -Vp is the negative of the Poisson bracket operator, and the subscript W indicates the partial Wigner transform. The partial Wigner transform of the total Hamiltonian is written as,... 

The last line defines the mixed quantum-classical Liouville operator C. The W subscripts denote a partial Wigner transform of an operator or density matrix. The phase space variables of the bath are (R,P) and the partial Wigner transform of the total hamiltonian is given by,... 

The last expression allows us to write the partial Wigner transform of the hamiltonian... 

The subscript W refers to this partial Wigner transform, N is the eoordinate space dimension of the bath and X = R, P). In this partial Wigner representation, the Hamiltonian of the system takes the form Hw R,P) = P /2M + y-/2m+ V q,R). If the subsystem DOF are represented using the states of an adiabatic basis, a P), which are the solutions of hw R) I R)=Ea R) I where hw K)=p /2m+ V q,R) is the Hamiltonian for the subsystem with fixed eoordinates R of the bath, the density matrix elements are p i -, 0 = ( I Pw( 01 )- From the solution of the quantum Liouville equation given some initial state of the entire quantum system, the reduced density matrix elements of the quantum subsystem of interest can be obtained by integrating over the bath variables, p f t) = dX p X,t), in order to find the populations and off-diagonal elements (coherences) of the density matrix. 

We have used the fact that the integration in this matrix element runs over electronic coordinates, and does not affect the nuclear coordinates. The Wigner-Eckart theorem can be applied to derive the selection rules. Since the Hamiltonian is invariant under the elements of the symmetry group, the transformation properties of the operator part in this matrix element will be determined by the partial derivatives, d/dQry-Aswt have seen in Sect. 1.3, a partial derivative in a variable has the same transformation properties as the variable itself. The operator part is thus given by ... 

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