11.12 Linear-Scaling Computation of Electric Properties

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, $𝓔$:

$${\widehat{H}}_{\mathrm{field}}=\frac{1}{2}\widehat{𝝁}\mathbf{\cdot }𝓔(e^{-i\omega t}+e^{+i\omega t})$$

(11.68)

with

$$\widehat{𝝁}=-e\sum _i^{N_{\mathrm{elec}}}{\widehat{𝐫}}_i.$$

(11.69)

Here, $\omega$ is the field frequency and $\widehat{𝝁}$ is the dipole moment operator. The TDSCF equations can be solved via standard techniques of perturbation theory.⁸³¹ As a solution, one obtains the first-order perturbed density matrix [${𝐏}^x(\pm \omega )$] and the second-order perturbed density matrices [${𝐏}^{xy}(\pm \omega ,\pm {\omega }^{\prime })$]. From these quantities, the following properties can be calculated:

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  Static polarizability: ${\alpha }_{xy}(0;0)=\mathrm{tr}\left[{𝐇}^{{\mu }_x}{𝐏}^y(\omega =0)\right]$

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  Dynamic polarizability: ${\alpha }_{xy}(\pm \omega ;\mp \omega )=\mathrm{tr}\left[{𝐇}^{{\mu }_x}{𝐏}^y(\pm \omega )\right]$

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  Static hyperpolarizability: ${\beta }_{xyz}(0;0,0)=\mathrm{tr}\left[{𝐇}^{{\mu }_x}{𝐏}^{yz}(\omega =0,\omega =0)\right]$

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  Second harmonic generation: ${\beta }_{xyz}(\mp 2\omega ;\pm \omega ,\pm \omega )=\mathrm{tr}\left[{𝐇}^{{\mu }_x}{𝐏}^{yz}(\pm \omega ,\pm \omega )\right]$

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  Electro-optical Pockels effect: ${\beta }_{\mathrm{xyz}}(\mp \omega ;0,\pm \omega )=\mathrm{tr}\left[{𝐇}^{{\mu }_x}{𝐏}^{yz}(\omega =0,\pm \omega )\right]$

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  Optical rectification: ${\beta }_{xyz}(0;\pm \omega ,\mp \omega )=\mathrm{tr}\left[{𝐇}^{{\mu }_x}{𝐏}^{yz}(\pm \omega ,\mp \omega )\right]$

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

Note that third-order properties (${\beta }_{xyz}$) can be computed either with the equations above, which is based on a second-order TDSCF calculation (for ${𝐏}^{yz}$), or alternatively from first-order properties using Wigner’s $2n+1$ rule.⁴⁵⁴ The second-order approach corresponds to MOPROP job numbers 101 and 102 (see below) whereas use of the $2n+1$ rule corresponds to job numbers 103 and 104. Solution of the second-order TDSCF equations depends upon first-order results and therefore convergence can be more problematic as compared to the first-order calculation. For this reason, we recommend job numbers 103 and 104 for the calculation of first hyperpolarizabilities.

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 $𝒪(N^3)$. 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 $𝒪(N^2)$ scaling but with a very large prefactor. The cost of these integral contractions can be reduced from quadratic to $𝒪(N)$ using LinK/CFMM methods (Section 4.6).⁵¹⁴ All derivatives are computed analytically.

The TDSCF module in Q-Chem is know as “MOProp", since it corresponds (formally) to time propagation of the molecular orbitals. (For actual time propagation of the MOs, see Section 7.12.) The MOProp module has the following features:

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

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  Analytic derivatives

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  DIIS acceleration

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

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  Support for LDA, GGA, and global hybrid functionals. Meta-GGA and range-separated functionals are not yet supported, nor are functionals that contain non-local correlation (e.g., those containing VV10).