Quantum Free Energy Calculations

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a 

For a system at temperature T = l/k-Qp in the potential V(x) and having the Hamiltonian operator H = K+V = —(fi2/2m)V2+l/(x), the elements of the density matrix operator 1H in the canonical ensemble are defined, in the coordinate representation, as 

These are Green s functions diffusing in what is interpreted as an imaginary time [3, according to the Bloch equation, dp/8(3 = JYp (a diffusion-type partial differential equation). These Green s functions satisfy the equation 

For large P, (3/P is small and it is possible to find a good short-time approximation to the Green function p. This is usually done by employing the Trotter product formula for the exponentials of the noncommuting operators K and V 

In the so-called primitive representation of the discretized path-integral approach [141], the canonical partition function for finite P has the form 

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Enhanced sampling in conformational space is not only relevant to sampling classical degrees of freedom. An additional reason to illustrate this particular method is that the delocalization feature of the underlying distribution in Tsallis statistics is useful to accelerate convergence of calculations in quantum thermodynamics [34], We focus on a related method that enhances sampling for quantum free energies in Sect. 8.4.2. 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

The Feynman-Hibbs and QFH models perform quite well in free energy calculations as long as the quantum corrections are modest. The conditions for validity of the approximations are given above. 

We can expect to see future research directed at QM/MM and ab initio simulation methods to handle these electronic structure effects coupled with path integral or approximate quantum free energy methods to treat nuclear quantum effects. These topics are broadly reviewed in [32], Nuclear quantum effects for the proton in water have already received some attention [30, 76, 77]. Utilizing the various methods briefly described above (and other related approaches), free energy calculations have been performed for a wide range of problems involving proton motion [30, 67-69, 71, 72, 78-80]. 

Miller, T. F., Ill Clary, D. C., Torsional path integral Monte Carlo method for calculating the absolute quantum free energy of large molecules, J. Chem. Phys. 2003,119, 68-76... 

A QUANTUM CHEMICAL APPROACH TO FREE ENERGY CALCULATION FOR CHEMICAL REACTIONS IN CONDENSED SYSTEM COMBINATION OF A QUANTUM CHEMICAL METHOD WITH A THEORY OF STATISTICAL MECHANICS... 

A Quantum Chemical Approach to Free Energy Calculation... 

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