Thermodynamic integration with harmonic reference

The Helmholtz free energy ($A$) of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (0,$\mathbf{x}$) as follows

$A_{1} = A_{0,\mathbf{x}} + \Delta A_{0,\mathbf{x}\rightarrow 1}$

where $\Delta A_{0,\mathbf{x}\rightarrow 1}$ is anharmonic free energy. The latter term can be determined by means of thermodynamic integration (TI)

$\Delta A_{0,\mathbf{x}\rightarrow 1} = \int_0^1 d\lambda \langle V_1 -V_{0,\mathbf{x}} \rangle_\lambda$

with $V_i$ being the potential energy of system $i$, $\lambda$ is a coupling constant and $\langle\cdots\rangle_\lambda$ is the NVT ensemble average of the system driven by the Hamiltonian

$\mathcal{H}_\lambda = \lambda \mathcal{H}_1 + (1-\lambda)\mathcal{H}_{0,\mathbf{x}}$

Free energy of harmonic reference system within the quasi-classical theory writes

$A_{0,\mathbf{x}} = A_\mathrm{el}(\mathbf{x}_0) - k_\mathrm{B} T \sum_{i = 1}^{N_\mathrm{vib}} \ln \frac{k_\mathrm{B} T}{\hbar \omega_i}$

with the electronic free energy $A_\mathrm{el}(\mathbf{x}_0)$ for the configuration corresponding to the potential energy minimum with the atomic position vector $\mathbf{x}_0$, the number of vibrational degrees of freedom $N_\mathrm{vib}$, and the angular frequency $\omega_i$ of vibrational mode $i$ obtained using the Hesse matrix $\underline{\mathbf{H}}^\mathbf{x}$. Finally, the harmonic potential energy is expressed as

$V_{0,\mathbf{x}}(\mathbf{x}) = V_{0,\mathbf{x}}(\mathbf{x}_0) + \frac{1}{2} (\mathbf{x} - \mathbf{x}_0)^T \underline{\mathbf{H}}^\mathbf{x} (\mathbf{x} - \mathbf{x}_0)$

Thus, a conventional TI calculation consists of the following steps:

1. determine $\mathbf{x}_0$ and $V_{0,\mathbf{x}}(\mathbf{x}_0)$ in structural relaxation
2. compute $\omega_i$ in vibrational analysis
3. use the data obtained in the point 2 to determine $\underline{\mathbf{H}}^\mathbf{x}$ that defines the harmonic forcefield
4. perform NVT MD simulations for several values of $\lambda \in \langle0,1\rangle$ and determine $\langle V_1 -V_{0,\mathbf{x}} \rangle$
5. integrate $\langle V_1 -V_{0,\mathbf{x}} \rangle$ over the $\lambda$ grid and compute $\Delta A_{0,\mathbf{x}\rightarrow 1}$

Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because $V_{0,\mathbf{x}}(\mathbf{x})$ is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve position of the center of mass and is therefore unsuitable for the use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with $\lambda \rightarrow$ 0 is likely to generate unphysical configurations causing serious convergence issues. These problems have been addressed in series of works by Amsler et al.

First, the method was formulated in terms of rotationally and translationally invariant internal coordinates $\mathbf{q}=\mathbf{q}(\mathbf{x})$, whereby the free energy of interacting system is repartitioned as follows:

$A_1 = A_{0,\mathbf{x}} + \Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}} + \Delta A_{0,\mathbf{q} \rightarrow 1}$

where $\Delta A_{0,\mathbf{x} \rightarrow 0,\mathbf{q}}$ is the free energy change due to transformation from the system harmonic in $\mathbf{x}$ into the system harmonic in $\mathbf{q}$ and $\Delta A_{0,\mathbf{q} \rightarrow 1}$ is that for the transformation of the latter into a fully interacting system. The force field for the system harmonic in $\mathbf{q}$ is defined as:

$V_{0,\mathbf{q}}(\mathbf{q}) = V_{0,\mathbf{q}}(\mathbf{q}_0) + \frac{1}{2} (\mathbf{q} - \mathbf{q}_0)^T \mathbf{\underline{H}^q} (\mathbf{q} - \mathbf{q}_0)$

where $\mathbf{q}_0=\mathbf{q}(\mathbf{x}_0)$ and the Hesse matrix $\mathbf{\underline{H}}^\mathbf{q}$ defined for a potential energy minimum is related to $\mathbf{\underline{H}}^\mathbf{x}$ via $\mathbf{\underline{H}}^\mathbf{x} = \mathbf{\underline{B}}^T \mathbf{\underline{H}}^\mathbf{q} \mathbf{\underline{B}}$ with $\mathbf{\underline{B}}_{i,j} = \frac{\partial q_i}{\partial x_j}$ being the Wilson matrix.