Quasi-Harmonic Approximation

The quasi-harmonic approximation (QHA) extends harmonic lattice dynamics by allowing the phonon frequencies to depend on crystal volume. At each fixed volume the crystal is still treated harmonically, but the collection of harmonic calculations defines a temperature-dependent free-energy surface from which equilibrium properties can be obtained at finite pressure and temperature [1] [2].

The treatment in this section describes the scientific model. The numerical representations, fit diagnostics, and execution choices available in Quantas are documented in Quasi-Harmonic Approximation: implementation and workflow.

Why the harmonic approximation is insufficient

In the harmonic approximation the vibrational frequencies are evaluated for a fixed structure. The equilibrium volume therefore does not change with temperature, which implies

\[\alpha_V = 0, \qquad C_P = C_V.\]

Real crystals expand or contract because their vibrational spectrum changes with volume. The QHA introduces this effect while retaining a harmonic phonon description at each sampled volume.

Quasi-harmonic free-energy surface

Let \(U_0(V)\) be the static electronic energy and let \(F_{\mathrm{vib}}(V,T)\) be the harmonic vibrational Helmholtz free energy calculated from the volume-dependent phonon frequencies. The total Helmholtz free energy is

\[F(V,T) = U_0(V) + F_{\mathrm{vib}}(V,T).\]

The volume dependence of \(F_{\mathrm{vib}}\) is the essential new element of the QHA. It describes the leading thermodynamic consequence of vibrational anharmonicity without introducing explicit cubic or higher-order force constants [3].

Equilibrium at finite pressure

At an external hydrostatic pressure \(P\), the equilibrium state minimizes

\[G^*(V;P,T) = F(V,T) + PV.\]

The equilibrium volume \(V(P,T)\) satisfies

\[\left(\frac{\partial G^*}{\partial V}\right)_{P,T} = 0,\]

or equivalently

\[P = -\left(\frac{\partial F}{\partial V}\right)_T.\]

Once the equilibrium volume has been determined, harmonic thermodynamic properties are evaluated or interpolated at that volume to construct the pressure–temperature state.

Thermodynamic potentials

At the equilibrium volume, the enthalpy and Gibbs free energy are

\[H(P,T) = U(P,T) + PV,\]

and

\[G(P,T) = F(P,T) + PV.\]

The Gibbs free energy is the appropriate thermodynamic potential for comparing phases at fixed pressure and temperature, provided all phases are represented with mutually consistent electronic and vibrational free energies.

Isothermal bulk modulus

The isothermal bulk modulus is obtained from the curvature of the Helmholtz free-energy surface,

\[K_T(V,T) = V\left(\frac{\partial^2 F}{\partial V^2}\right)_T = -V\left(\frac{\partial P}{\partial V}\right)_T.\]

At equilibrium it becomes a function of pressure and temperature, \(K_T(P,T)\). Its pressure derivative is

\[K'_T = \left(\frac{\partial K_T}{\partial P}\right)_T.\]

A positive \(K_T\) is required for local mechanical stability against hydrostatic volume fluctuations.

Thermal expansion

The volumetric thermal-expansion coefficient is

\[\alpha_V(P,T) = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P.\]

Using

\[P(V,T) = -\left(\frac{\partial F}{\partial V}\right)_T,\]

and differentiating the constant-pressure condition gives the mixed-derivative form

\[\alpha_V = -\frac{1}{K_T} \frac{\partial^2 F}{\partial V\,\partial T}.\]

Since

\[S = -\left(\frac{\partial F}{\partial T}\right)_V,\]

this can also be written as the Maxwell-relation form

\[\alpha_V = \frac{1}{K_T} \left(\frac{\partial S}{\partial V}\right)_T.\]

This relation is especially useful because it connects thermal expansion to the volume dependence of the vibrational entropy.

Mode Grüneisen parameters

The volume sensitivity of a phonon mode is described by its mode Grüneisen parameter,

\[\gamma_{qj}(V) = -\frac{\partial\ln\nu_{qj}(V)} {\partial\ln V}.\]

A positive mode Grüneisen parameter means that the mode softens upon expansion; a negative value means that the mode stiffens upon expansion.

A heat-capacity-weighted average is

\[\bar{\gamma}(V,T) = \frac{ \sum_{qj} w_q\,\gamma_{qj}(V) C_{V,qj}(V,T) }{ \sum_{qj} w_q\,C_{V,qj}(V,T) }.\]

When the quasi-harmonic relations are internally consistent, the volumetric thermal expansion can be expressed as

\[\alpha_V = \frac{\bar{\gamma} C_V}{K_T V},\]

with all quantities expressed on the same normalization basis. Conversely, the macroscopic thermodynamic Grüneisen parameter is

\[\gamma = \frac{\alpha_V K_T V}{C_V}.\]

The mode-resolved route requires physically continuous phonon branches across the sampled volumes. The macroscopic thermodynamic relation does not require mode-by-mode tracking, but it remains sensitive to the quality of \(\alpha_V\), \(K_T\), and \(C_V\).

Isochoric and isobaric heat capacities

The isochoric heat capacity is obtained from harmonic thermodynamics evaluated at the equilibrium volume. The standard thermodynamic relation between the isobaric and isochoric heat capacities is

\[C_P-C_V = \alpha_V^2 K_T V T.\]

Therefore

\[C_P = C_V + \alpha_V^2 K_T V T.\]

The same normalization and unit basis must be used for \(C_V\), \(K_T\), and \(V\).

Adiabatic bulk modulus

The adiabatic and isothermal bulk moduli are related by

\[\frac{K_S}{K_T} = \frac{C_P}{C_V},\]

so that

\[K_S = K_T\frac{C_P}{C_V}.\]

At zero temperature, or whenever the thermal correction vanishes, \(K_S\) approaches \(K_T\).

Structural properties

When a consistent set of relaxed crystal structures is available along the sampled volume path, the equilibrium cell can be reconstructed at \(V(P,T)\). The linear expansion of a lattice parameter \(a\) can be written through the chain rule as

\[\alpha_a = \frac{1}{a}\left(\frac{\partial a}{\partial T}\right)_P = \frac{\partial\ln a}{\partial\ln V}\,\alpha_V.\]

Equivalent relations apply to the other lattice parameters and to the symmetric Cartesian thermal-expansion tensor. The trace of that tensor must be consistent with the volumetric expansion coefficient,

\[\operatorname{tr}(\boldsymbol{\alpha}) = \alpha_V.\]

This structural reconstruction represents the cell-shape response along the sampled volume-constrained structural path. It is not a fully independent anisotropic minimization of every strain degree of freedom at every pressure–temperature point.

Two equivalent QHA viewpoints

The QHA can be organized in two scientifically equivalent ways when the underlying fits are accurate.

Frequency-based formulation

The phonon frequencies are represented as functions of volume. Harmonic thermodynamic functions are then recalculated at the equilibrium volume. This formulation retains mode-resolved information and permits direct calculation of mode Grüneisen parameters, but requires mode continuity.

Thermodynamic formulation

Harmonic thermodynamic quantities such as \(F_{\mathrm{vib}}\), \(S\), and \(C_V\) are first calculated at the sampled volumes and then represented as functions of volume. This avoids mode-by-mode correspondence, but does not retain the same mode-resolved information.

Quantas supports both formulations. Their numerical configuration belongs to the workflow layer rather than to the scientific definition of the QHA.

Scientific scope and limitations

The QHA captures the leading thermal effect arising from the volume dependence of harmonic phonons. It does not include explicit anharmonic interactions at fixed volume. Important limitations therefore include:

• phonon linewidths and lifetimes are not described;

• intrinsic temperature shifts of frequencies at fixed volume are neglected;

• strongly anharmonic crystals may not be represented accurately;

• dynamically unstable modes and soft-mode phase transitions require special treatment;

• results become unreliable when the equilibrium state lies far outside the sampled volume interval;

• electronic, magnetic, configurational, or defect contributions must be added separately when they are thermodynamically relevant.

The QHA is generally most reliable for mechanically stable crystalline phases away from melting, strongly anharmonic regimes, and structural phase transitions. Its quality must be assessed from the phonon calculations, the sampled volume range, the free-energy representation, and comparison with experimental or independent theoretical data [4].

Quantities derived by Quantas

Depending on the supplied data and selected analysis, the QHA workflow can provide:

• equilibrium volume \(V(P,T)\);

• zero-point, thermal, internal, Helmholtz, enthalpy, and Gibbs energies;

• entropy \(S(P,T)\);

• \(C_V(P,T)\) and \(C_P(P,T)\);

• \(K_T(P,T)\), \(K_S(P,T)\), and \(K'_T(P,T)\);

• volumetric thermal expansion \(\alpha_V(P,T)\);

• thermodynamic and mode-weighted Grüneisen parameters;

• mode-resolved Grüneisen parameters when branch continuity is available;

• equilibrium lattice parameters, axial expansion coefficients, and the thermal-expansion tensor when a structural volume path is supplied.

The practical sequence for obtaining and validating these quantities is described in Quasi-Harmonic Approximation: implementation and workflow.

Bibliographic references

[1]

Orson L. Anderson. Equations of State of Solids for Geophysics and Ceramic Science. Oxford University Press, New York (1995).

[2]

Orson L. Anderson, Koji Masuda, and Donald G. Isaak. “A new thermodynamic approach for high-pressure physics”. Physics of the Earth and Planetary Interiors, 91, 3-16 (1995). DOI: 10.1016/0031-9201(95)03044-W.

[3]

Alessandro Erba. “On combining temperature and pressure effects on structural properties of crystals with standard ab initio techniques”. Journal of Chemical Physics, 141, 124115 (2014). DOI: 10.1063/1.4896228.

[4]

Alessandro Erba, M. Shahrokhi, R. Moradian, and Roberto Dovesi. “On how differently the quasi-harmonic approximation works for two isostructural crystals: Thermal properties of periclase and lime”. Journal of Chemical Physics, 142, 044114 (2015). DOI: 10.1063/1.4906422.