This article reviews the current status of lattice-dynamical calculations in crystals, using density-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calculation of the response to macroscopic electric fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodology is demonstrated with a number of applications existing in the literature.

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Phonons and related crystal properties from density-functional perturbation theory

A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initioMO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initioSCF–MO calculations.

Optimization of equilibrium geometries and transition structures

https://th.fhi-berlin.mpg.de/site/uploads/Publications/CPC-180-2175-2009.pdf

Ab Initio Molecular Simulations with Numeric Atom-Centered Orbitals

The general expression for the exact forces on the nuclei (negative derivatives of the total energy with respect to the nuclear coordinates) is applied for Hartree-Fock wavefunctions. It is suggested that force constants should be calculated by differentiating the forces numerically. This method, called the force method, is numerically more accurate and requires less computation than the customary one of differentiating the energy numerically twice. It permits the quick determination of the equilibrium geometry by relaxing the nuclear coordinates until the forces vanish. The unreliability of the methods using the Hellmann-Feynman forces is re-emphasized. The question of which force constants can be best calculated ab initio is discussed.

Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules

A new, noniterative triples correction to the coupled‐cluster singles and doubles (CCSD), method, for general single determinant reference functions is proposed and investigated numerically for various cases, including non‐Hartree–Fock (non‐HF) reference functions. It is correct through fourth‐order of perturbation theory for non‐HF references, and unlike other such methods, retains the usual invariance properties common to CC methods, while requiring only a single N7 step. In the canonical Hartree–Fock case, the method is equivalent to the usual CCSD(T) method, but now permits the use of restricted open‐shell Hartree‐Fock (ROHF) and quasirestricted Hartree–Fock (QRHF) reference determinants, along with many others. Comparisons with full configuration interaction (FCI) results are presented for CH2, CH2+, CH3, NH2, and SiH2. The paper also reports the derivation and initial computational implementation of analytical gradients for the ROHF‐CCSD(T) method, which includes unrestricted Hartree–Fock (UHF) CCSD...

Coupled‐cluster methods with noniterative triple excitations for restricted open‐shell Hartree–Fock and other general single determinant reference functions. Energies and analytical gradients

General expressions for the force constants and dipole‐moment derivatives of molecules are derived, and the problems arising in their practical application are reviewed. Great emphasis is placed on the use of the Hartree–Fock function as an approximate wavefunction, and a number of its properties are discussed and re‐emphasised. The main content of this paper is the development of a perturbed Hartree–Fock theory that makes possible the direct calculation of force constants and dipole‐moment derivatives from SCF–MO wavefunctions. Essentially the theory yields ∂φi / ∂RJα, the derivative of an MO with respect to a nuclear coordinate.

Force Constants and Dipole-Moment Derivatives of Molecules from Perturbed Hartree-Fock Calculations. I

The perturbed Hartree–Fock theory developed in the preceding paper is applied to LiH, BH, and HF, using limited basis‐set SCF–MO wavefunctions derived by previous workers. The calculated values for the force constant ke and the dipole‐moment derivative μ(1) are (experimental values in parentheses): LiH, ke = 1.618(1.026) mdyn/A, μ(1) = −18.77(−2.0±0.3) D/A BH, ke = 5.199(3.032) mdyn/A, μ(1) = −1.03(−)D/A; HF, ke = 12.90(9.651) mdyn/A, μ(1) = −2.15(+1.50) D/A. The values of the force on the proton were calculated exactly and according to the Hellmann–Feynman theorem in each case, and the discrepancies show that none of the wavefunctions used are close to the Hartree–Fock limit, so that the large errors in ke and μ(1) are not surprising. However no difficulties arose in the perturbed Hartree–Fock calculation, so that the application of the theory to more accurate wavefunctions appears quite feasible.

Force constants and dipole moment derivatives of molecules from perturbed Hartree-Fock calculations II. Applications to limited basis set SCF-MO wavefunctions

J. Gerratt

Papers

Force Constants and Dipole-Moment Derivatives of Molecules from Perturbed Hartree-Fock Calculations. I

Force constants and dipole moment derivatives of molecules from perturbed Hartree-Fock calculations II. Applications to limited basis set SCF-MO wavefunctions