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# 10.10.1 Introduction

(February 4, 2022)

The search for new optical devices is a major field of materials sciences. Here, polarizabilities and hyperpolarizabilities provide particularly important information on molecular systems. The response of the molecular systems in the presence of an external, monochromatic, oscillatory electric field is determined by the solution of the time-dependent SCF (TDSCF) equations. Within the dipole approximation, the perturbation is represented as the interaction of the molecule with a single Fourier component of the external field, $\bm{{\cal E}}$:

 $\hat{H}_{\rm field}=\tfrac{1}{2}\hat{\bm{\mu}}\bm{\cdot}\bm{{\cal E}}(e^{-i% \omega t}+e^{+i\omega t})$ (10.63)

with

 $\hat{\bm{\mu}}=-e\sum_{i}^{N_{\rm elec}}\hat{\mathbf{r}}_{i}\;.$ (10.64)

Here, $\omega$ is the field frequency and $\hat{\bm{\mu}}$ is the dipole moment operator. The TDSCF equations can be solved via standard techniques of perturbation theory.1016 As a solution, one obtains the first-order perturbed density matrix [$\mathbf{P}^{x}(\pm\omega)$] and the second-order perturbed density matrices [$\mathbf{P}^{xy}(\pm\omega,\pm\omega^{\prime})$]. From these quantities, the following properties can be calculated:

• Static polarizability: $\alpha_{xy}(0;0)=\mathrm{tr}\bigl{[}\mathbf{H}^{\mu_{x}}\mathbf{P}^{y}(\omega=% 0)\bigr{]}$

• Dynamic polarizability: $\alpha_{xy}(\pm\omega;\mp\omega)=\mathrm{tr}\bigl{[}\mathbf{H}^{\mu_{x}}% \mathbf{P}^{y}(\pm\omega)\bigr{]}$

• Static hyperpolarizability: $\beta_{xyz}(0;0,0)=\mathrm{tr}\bigl{[}\mathbf{H}^{\mu_{x}}\mathbf{P}^{yz}(% \omega=0,\omega=0)\bigr{]}$

• Second harmonic generation: $\beta_{xyz}(\mp 2\omega;\pm\omega,\pm\omega)=\mathrm{tr}\bigl{[}\mathbf{H}^{% \mu_{x}}\mathbf{P}^{yz}(\pm\omega,\pm\omega)\bigr{]}$

• Electro-optical Pockels effect: $\beta_{\mathrm{xyz}}(\mp\omega;0,\pm\omega)=\mathrm{tr}\bigl{[}\mathbf{H}^{\mu% _{x}}\mathbf{P}^{yz}(\omega=0,\pm\omega)\bigr{]}$

• Optical rectification: $\beta_{xyz}(0;\pm\omega,\mp\omega)=\mathrm{tr}\bigl{[}\mathbf{H}^{\mu_{x}}% \mathbf{P}^{yz}(\pm\omega,\mp\omega)\bigr{]}$

Here, $\mathbf{H}^{\mu_{x}}$ is the matrix representation of the $x$ component of the dipole moment.

For third-order properties ($\beta_{xyz}$), rather than computing them using a second-order TDSCF calculation and solving for $\mathbf{P}^{yz}$ explicitly, we calculate them from first-order properties using Wigner’s $2n+1$ rule.549

The TDSCF calculation is more time-consuming than the SCF calculation that precedes it (where the field-free, unperturbed ground state of the molecule is obtained). Q-Chem’s implementation of the TDSCF equations is MO based and the cost therefore formally scales asymptotically as ${\cal{O}}({N^{3}})$. The prefactor of the cubic-scaling step is rather small, however, and in practice (over a wide range of molecular sizes) the calculation is dominated by the cost of contractions with two-electron integrals, which is formally ${\cal{O}}({N^{2}})$ scaling but with a very large prefactor. The cost of these integral contractions can be reduced from quadratic to ${\cal{O}}({N})$ using LinK/CFMM methods (Section 4.6).614 All derivatives are computed analytically.

The TDSCF module in Q-Chem is known as “MOProp”, since it corresponds (formally) to time propagation of the molecular orbitals. (For actual time propagation of the MOs, see Section 7.4.) The MOProp module has the following features:

• LinK and CFMM support to evaluate Coulomb- and exchange-like matrices

• Analytic derivatives

• DIIS acceleration

• Both restricted and unrestricted implementations of CPSCF and TDSCF equations are available, for both Hartree-Fock and Kohn-Sham DFT.

• Support for LDA, GGA, Meta-GGA763, global hybrid and common range-separated functionals. VV10 is the only non-local correlation functional supported.