Draft:Vibrational Configuration Interaction

The term configuration as occurring in VCI refers to the simplest description of a 3N−6 dimensional vibrational wavefunction, ${\displaystyle |\Phi ^{I}\rangle }$, by means of a product of 3N−6 one-dimensional wavefunctions, ${\displaystyle |\varphi _{i}^{I}\rangle }$. This product is called a Hartree product, which can be written as:

${\displaystyle |\Phi ^{I}\rangle =\prod _{i}^{3N-6}|\varphi _{i}^{I}\rangle }$

The one-dimensional wavefunctions within a Hartree product can be either harmonic oscillator functions, the solutions of vibrational self-consistent field (VSCF) calculations, or any other basis functions.

Within VCI theory a vibrational state is given by a linear combination of configurations, that is

${\displaystyle |\Psi _{v}\rangle =\sum _{I}C_{vI}|\Phi ^{I}\rangle }$

As the number of configurations grows rapidly and becomes intractably large, the linear expansion is structured with respect to the excitation levels (in terms of quantum numbers) of the configurations. Thus the vibrational wavefunction can be expressed as

${\displaystyle |\Psi _{v}\rangle =C_{v0}+\sum _{S}C_{vS}|S\rangle +\sum _{D}C_{vD}|D\rangle +\sum _{T}C_{vT}|T\rangle +\dots }$

Here, ${\displaystyle |S\rangle }$, ${\displaystyle |D\rangle }$, ${\displaystyle |T\rangle }$ denote singly, doubly and triply excited configurations. For molecules with more than 3 or 4 atoms, this expansion needs to be truncated after a certain excitation level.

The molecular Hamilton operator for solving the nuclear Schrödinger equation depends on the coordinate system being employed. Most programs are based on the Watson Hamiltonian, the Podolsky Hamiltonian or molecule-specific Hamiltonians. The representation of the wavefunction by a linear combination of configurations allows one to transform the nuclear Schrödinger equation into a matrix eigenvalue problem, i.e.

${\displaystyle \mathbf {H} \mathbf {C} =\mathbf {C} \mathbf {E} }$

with ${\displaystyle \mathbf {E} }$ denoting the diagonal matrix of eigenvalues (vibrational state energies), and the eigenvectors ${\displaystyle \mathbf {C} }$ yielding the coefficients of the vibrational wavefunction.

For small systems, the solutions of the VCI eigenvalue problem yield exact results within the chosen basis of one-dimensional functions, which is termed full VCI (FVCI). However, VCI theory suffers from two computational bottlenecks: the high dimensionality of the Hamilton operator and the huge number of possible configurations, which leads to truncations and thus approximate solutions. As a result, the Hamiltonian is typically represented by a truncated series expansion, e.g. Taylor expansion or n-mode expansion.

The truncation of the Hamiltonian and the restriction of configurations to low excitation levels leads to the size-extensivity problem. However, once high-order expansion terms are included, the final results approach the FVCI limit and the size-extensivity error becomes small compared to other error sources, such as the quality of the electronic structure method used to generate the potential energy surface. Depending on many parameters, VCI theory generally yields very accurate results and can be systematically improved by including higher-order terms.

In principle, all eigenvalues relevant for a vibrational spectrum can be obtained from a single VCI matrix and its diagonalization. In contrast, state-specific VCI calculations are tailored to individual vibrational states. In these calculations, state-specific one-dimensional wavefunctions can be used and the correlation space can be restricted to configurations relevant for the targeted state. This leads to so-called configuration-selective VCI calculations, in which important configurations are selected based on a given criterion. This results in a significant reduction of the configuration space and thus substantial savings in computational cost.

For the calculation of a full vibrational spectrum, this approach requires many smaller VCI calculations. A formal drawback of this variant is the non-orthogonality of the resulting wavefunctions, but in practice this effect is usually very small and can be neglected.

• MULTIMODE: First normal mode based software for VSCF/VCI calculations introduced by Joel M. Bowman. ^[5]. Implementation available at Github. ^[6].
• Q-Chem: A general-purpose quantum chemistry software package including implementations of VCI methods, as introduced by R. J. Whitehead and N. C. Handy. ^[7] The implementation is described in the Q-Chem manual. ^[8]
• PyVCI: Python-based implementation of VSCF/VCI methods developed by M. Sibaev and D. L. Crittenden. ^[9] The implementation is available as open-source software. ^[10]

• MOLPRO: A general purpose quantum chemistry software package that includes a versatile PES generator and VSCF/VCI modules. ^[11] The modules were introduced by G. Rauhut. ^[12]

• CRYSTAL: A periodic quantum chemistry software package implementing VSCF and VCI methods for crystalline systems. The implementation was described by A. Erba et al. ^[13] The relevant sections are documented in the CRYSTAL23 manual, chapter 8.12. ^[14]

• VHCI: Vibrational Heat-Bath Configuration Interaction approach introduced by J. H. Fetherolf and T. C. Berkelbach. ^[15] Open-source implementations are available. ^{[16] [17]}