## Friday, November 26, 2021

### Quantum harmonic free energies for biomolecules and nanomaterials

Figure 2 from the paper. (c) The authors. Reproduced under the CC-BY license.

This paper describes a method by which the harmonic vibrational free energy contributions can be accurately approximated at roughly 10% of the cost of a conventional Hessian calculation.

The equations for the vibrational free energy contributions are recast in terms of the trace of a matrix function (remember that the trace of a matrix is equal to the sum of its eigenvalues). This removes the need for matrix diagonalisation, which is costly for large matrices. Then they use a stochastic estimator of the trace where the trace is rewritten in terms of displacements along $n$ random vectors. The accuracy of free energy differences can be further increased by using the same random vectors for both reactants and products.

The accuracy of this approximation increases with the number of displacement vectors (and, hence, gradient evaluations) used. The authors tested in one several large systems, such as protein-ligand binding,  and found that sub-kcal/mol accuracy can be obtained at about 10% of the cost of a conventional Hessian calculation plus diagonalisation.

It is now quite common to scale the entropy contributions from small (<100 cm$^{-1}$) frequencies to get better numerical stability. I am not sure whether this is possible in the current approach since individual frequencies are not computed explicitly.

The code and data is "available upon reasonable request" 😕