Generalization of the periodic LCAO approach in the CRYSTAL code to g-type orbitals

Abstract

The Linear Combination of Atomic Orbitals approach implemented into the public Crystal program for quantum-chemical simulations of n-dimensional periodic systems ($n = 0,1,2,3$) is here extended to g-type basis functions. A general algorithmic procedure is devised for the calculation of the coefficients needed for the analytical evaluation of one- and two-electron integrals for energy and forces within the recursive McMurchie–Davidson strategy, up to arbitrarily high quantum numbers. Explicit routines are generated for the calculation of all the coefficients needed for g-type functions, which ensure a very high computational efficiency. The code has been generalized in many respects so as to allow for the use of g-type functions for: (1) Hartree–Fock energy and forces; (2) density functional theory energy and forces (in either a local density, generalized gradient, meta-GGA or various hybrid approximations); (3) all-electron and pseudo-potential basis sets; (4) spin-restricted and unrestricted calculations; (5) coupled-perturbed Hartree–Fock/Kohn–Sham (hyper)-polarizability calculations; (6) projected density-of-states. The g-type basis functions are expected to play an important role in (1) the description of the electronic structure of heavy elements and in particular of lanthanides and actinides with occupied 4f and 5f bands, respectively, where they represent the first polarization, (2) those calculations requiring an accurate description of the electronic polarization.

Appendices

A Recursion relations at work

We show here how the recursion relations presented in Refs. [7] and [4] can be used to generate the E coefficients to arbitrarily high quantum numbers. The starting point for these relations is $E[0,0,0,0,0,0,0,0,0]=1$. Then, the recursion relations are used to generate the coefficients in the following order, where the bold quantum numbers indicate the ones being increased:

$$\begin{aligned}&(1) \ E[n,\mathbf{l },\mathbf{l },\tilde{n},0,0,t,u,v] \quad (2) \ E[n,0,0,\tilde{n},\tilde{\mathbf{l }},\tilde{\mathbf{l }},t,u,v]\\&(3) \ E[n,\mathbf{l },\mathbf{l },\tilde{n},\tilde{l},\tilde{l},t,u,v] \quad (4)\ E[n,\mathbf{l },0,\tilde{n},0,0,t,u,v] \\&(5)\ E[n,0,0,\tilde{n},\tilde{\mathbf{l }},0,t,u,v] \quad (6) \ E[n,\mathbf{l },0,\tilde{n},\tilde{l},0,t,u,v] \\&(7) \ E[n,\mathbf{l },\mathbf{l },\tilde{n},\tilde{l},0,t,u,v] \quad (8) \ E[n,0,0,\tilde{n},\tilde{\mathbf{l }},\tilde{\mathbf{m }},t,u,v]\\&(9) \ E[n,\mathbf{l },0,\tilde{n},\tilde{l},\tilde{m},t,u,v] \quad (10) \ E[n,\mathbf{l },\mathbf{l },\tilde{n},\tilde{l},\tilde{m},t,u,v] \\&(11) \ E[n,\mathbf{l },\mathbf{m },\tilde{n},\tilde{l},0,t,u,v] \quad (12) \ E[n,\mathbf{l },0,\tilde{n},\tilde{l},\tilde{l},t,u,v] \\&(13) \ E[n,\mathbf{l },\mathbf{m },\tilde{n},\tilde{l},\tilde{l},t,u,v] \quad (14) \ E[n,\mathbf{l },\mathbf{m },\tilde{n},\tilde{l},\tilde{m},t,u,v]\\&(15) \ E[\mathbf{n },l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v] \quad (16) \ E[n,l,m,\tilde{\mathbf{n }},\tilde{l},\tilde{m},t,u,v] \end{aligned}$$

At each step, the coefficients are generated for all of the $N^E(l,\tilde{l})$ triplets t, u, v which have potentially nonzero contributions.

B Formulae for analytical force coefficients

The calculation of the $G_k^K$ coefficients is similar to the procedure described in Appendix A, except that now the coefficients are zero for $t+u+v > 2n+2\tilde{n}+l+\tilde{l}+1$ and the starting point of the recurrences is as follows [8]:

$$\begin{aligned} G_k^A[0,0,0,0,0,0,0,0,0]&= 2\frac{\alpha \beta }{\gamma }(B_k-A_k) \times \\&\times E[0,0,0,0,0,0,0,0,0] \end{aligned}$$

and

$$\begin{aligned} G_k^A[0,0,0,0,0,0,\delta _{kx},\delta _{ky},\delta _{kz}] = \frac{\alpha }{\gamma }E[0,0,0,0,0,0,0,0,0] \end{aligned}$$

Doll et al. [8] provide formulae for increasing the quantum numbers on center A of the $G_x^A$; here we provide similar formulae for the $G_y^A$ and $G_z^A$ cases as well, which complete the formalism. The quantum numbers n and $\tilde{n}$ are omitted from the formulae in which they are constant.

1.1 B.1 Recurrence in n

1.1.1 B.1.1 Derivative in y

$$\begin{aligned}&G_y^A[n+1,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v]\nonumber \\&=\frac{1}{(2\gamma )^2}\bigg ( G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t-2,u,v]\nonumber \\&\quad +\,G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u-2,v] \nonumber \\&\quad +\, G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v-2] \bigg ) \nonumber \\&\quad +\,\frac{1}{\gamma } \bigg ( -E[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad +\, (P_x-A_x)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad \,+(P_y-A_y)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad \,+(P_z-A_z)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v-1] \bigg ) \nonumber \\&\quad +\, \Big ( |\mathbf P -\mathbf A |^2+\frac{1}{\gamma }\big ( t+u+v+\frac{3}{2} \big ) \Big ) \nonumber \\&\quad \times G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad -\,2(P_y-A_y)E[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,2(t+1)(P_x-A_x)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,2(u+1)(P_y-A_y)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,2(v+1)(P_z-A_z)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v+1] \nonumber \\&\quad -\,2(u+1)E[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,(t+2)(t+1)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t+2,u,v] \nonumber \\&\quad +\,(u+2)(u+1)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u+2,v] \nonumber \\&\quad +\,(v+2)(v+1)G_y^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v+2] \end{aligned}$$

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1.1.2 B 1.2 Derivative in z

$$\begin{aligned}&G_z^A[n+1,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v]\nonumber \\&=\frac{1}{(2\gamma )^2}\bigg ( G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t-2,u,v]\nonumber \\&\quad +\,G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u-2,v] \nonumber \\&\quad +\, G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v-2] \bigg ) \nonumber \\&\quad +\,\frac{1}{\gamma } \bigg ( -E[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v-1] \nonumber \\&\quad +\, (P_x-A_x)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad +\,(P_y-A_y)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad +\,(P_z-A_z)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v-1] \bigg ) \nonumber \\&\quad +\, \left( |\mathbf P -\mathbf A |^2+\frac{1}{\gamma }\big ( t+u+v+\frac{3}{2} \big ) \right) \nonumber \\&\quad \times G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad -\,2(P_z-A_z)E[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,2(t+1)(P_x-A_x)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,2(u+1)(P_y-A_y)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,2(v+1)(P_z-A_z)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v+1] \nonumber \\&\quad -\,2(v+1)E[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v+1] \nonumber \\&\quad +\,(t+2)(t+1)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t+2,u,v] \nonumber \\&\quad +\,(u+2)(u+1)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u+2,v] \nonumber \\&\quad +\,(v+2)(v+1)G_z^A[n,l,m,\tilde{n},\tilde{l},\tilde{m},t,u,v+2] \end{aligned}$$

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1.2 B.2 Recurrence in l and m

1.2.1 B.2.1 Derivative in y

$$\begin{aligned}&G_y^A[l+1,l+1,\tilde{l},\tilde{m},t,u,v]\nonumber \\&=(2l+1)\bigg (-iE[l,l,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,\frac{1}{2\gamma }G_y^A[l,l,\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad +\,(t+1)G_y^A[l,l,\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,(P_x-A_x)G_y^A[l,l,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,i\frac{1}{2\gamma }G_y^A[l,l,\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad +\,i(u+1)G_y^A[l,l,\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,i(P_y-A_y)G_y^A[l,l,\tilde{l},\tilde{m},t,u,v]\bigg ) \end{aligned}$$

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1.2.2 B.2.2 Derivative in z

$$\begin{aligned}&G_z^A[l+1,l+1,\tilde{l},\tilde{m},t,u,v]\nonumber \\&=(2l+1)\bigg (\frac{1}{2\gamma }G_z^A[l,l,\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad +\,(t+1)G_z^A[l,l,\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,(P_x-A_x)G_z^A[l,l,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,i\frac{1}{2\gamma }G_z^A[l,l,\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad +\,i(u+1)G_z^A[l,l,\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,i(P_y-A_y)G_z^A[l,l,\tilde{l},\tilde{m},t,u,v]\bigg ) \end{aligned}$$

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1.3 B.3 Recurrence in l and − m

1.3.1 B.3.1 Derivative in y

$$\begin{aligned}&G_y^A[l+1,-l-1,\tilde{l},\tilde{m},t,u,v] \nonumber \\&=(2l+1)\bigg (iE[l,-l,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,\frac{1}{2\gamma }G_y^A[l,-l,\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad +\,(t+1)G_y^A[l,-l,\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,(P_x-A_x)G_y^A[l,-l,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad -\,i\frac{1}{2\gamma }G_y^A[l,-l,\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad -\,i(u+1)G_y^A[l,-l,\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad -\,i(P_y-A_y)G_y^A[l,-l,\tilde{l},\tilde{m},t,u,v]\bigg ) \end{aligned}$$

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1.3.2 B.3.2 Derivative in z

$$\begin{aligned}&G_z^A[l+1,-l-1,\tilde{l},\tilde{m},t,u,v] \nonumber \\&=(2l+1)\bigg (\frac{1}{2\gamma }G_z^A[l,-l,\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad +\,(t+1)G_z^A[l,-l,\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,(P_x-A_x)G_z^A[l,-l,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad -\,i\frac{1}{2\gamma }G_z^A[l,-l,\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad -\,i(u+1)G_z^A[l,-l,\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad -\,i(P_y-A_y)G_z^A[l,-l,\tilde{l},\tilde{m},t,u,v]\bigg ) \end{aligned}$$

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1.4 B.4 Recurrence in l

1.4.1 B.4.1 Derivative in y

$$\begin{aligned}&G_y^A[l+1,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&=\frac{(2l+1)}{(l- |m |+1)} \bigg \{ \frac{1}{2\gamma }G_y^A[l,m,\tilde{l},\tilde{m},t,u,v-1] \nonumber \\&\quad +\,(P_z-A_z)G_y^A[l,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,(v+1)G_y^A[l,m,\tilde{l},\tilde{m},t,u,v+1] \bigg \} \nonumber \\&\quad -\,\frac{(l+|m |)}{(l- |m |+1)} \bigg \{ \frac{1}{(2\gamma )^2}\Big (G_y^A[l-1,m,\tilde{l},\tilde{m},t-2,u,v]\nonumber \\&\quad +\,G_y^A[l-1,m,\tilde{l},\tilde{m},t,u-2,v] \nonumber \\&\quad +\,G_y^A[l-1,m,\tilde{l},\tilde{m},t,u,v-2] \Big ) \nonumber \\&\quad +\,\frac{1}{\gamma } \Big ( (P_x-A_x)G_y^A[l-1,m,\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad -\, E[l-1,m,\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad +\,(P_y-A_y)G_y^A[l-1,m,\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad +\,(P_z-A_z)G_y^A[l-1,m,\tilde{l},\tilde{m},t,u,v-1]\Big ) \nonumber \\&\quad +\,\Big ( |\mathbf P -\mathbf A |^2+\frac{1}{\gamma }\big ( t+u+v+\frac{3}{2} \big ) \Big ) \nonumber \\&\quad \times G_y^A[l-1,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad -\,2(P_y-A_y)E[l-1,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,2(t+1)(P_x-A_x)G_y^A[l-1,m,\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,2(u+1)(P_y-A_y)G_y^A[l-1,m,\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,2(v+1)(P_z-A_z)G_y^A[l-1,m,\tilde{l},\tilde{m},t,u,v+1] \nonumber \\&\quad -\,2(u+1)E[l-1,m,\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,(t+2)(t+1)G_y^A[l-1,m,\tilde{l},\tilde{m},t+2,u,v] \nonumber \\&\quad +\,(u+2)(u+1)G_y^A[l-1,m,\tilde{l},\tilde{m},t,u+2,v] \nonumber \\&\quad +\,(v+2)(v+1)G_y^A[l-1,m,\tilde{l},\tilde{m},t,u,v+2] \bigg \} \end{aligned}$$

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1.4.2 B.4.2 Derivative in z

$$\begin{aligned}&G_z^A[l+1,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&=\frac{(2l+1)}{(l- |m |+1)} \bigg \{ \frac{1}{2\gamma }G_z^A[l,m,\tilde{l},\tilde{m},t,u,v-1] \nonumber \\&\quad -\,E[l,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,(P_z-A_z)G_z^A[l,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,(v+1)G_z^A[l,m,\tilde{l},\tilde{m},t,u,v+1] \bigg \} \nonumber \\&\quad -\,\frac{(l+|m |)}{(l- |m |+1)} \bigg \{ \frac{1}{(2\gamma )^2}\Big ( G_z^A[l-1,m,\tilde{l},\tilde{m},t-2,u,v] \nonumber \\&\quad +\,G_z^A[l-1,m,\tilde{l},\tilde{m},t,u-2,v] \nonumber \\&\quad +\,G_z^A[l-1,m,\tilde{l},\tilde{m},t,u,v-2] \Big ) \nonumber \\&\quad +\,\frac{1}{\gamma } \Big ( (P_x-A_x)G_z^A[l-1,m,\tilde{l},\tilde{m},t-1,u,v] \nonumber \\&\quad -\, E[l-1,m,\tilde{l},\tilde{m},t,u,v-1] \nonumber \\&\quad +\,(P_y-A_y)G_z^A[l-1,m,\tilde{l},\tilde{m},t,u-1,v] \nonumber \\&\quad +\,(P_z-A_z)G_z^A[l-1,m,\tilde{l},\tilde{m},t,u,v-1]\Big ) \nonumber \\&\quad +\,\Big ( |\mathbf P -\mathbf A |^2+\frac{1}{\gamma }\big ( t+u+v+\frac{3}{2} \big ) \Big ) \nonumber \\&\quad \times G_z^A[l-1,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad -\,2(P_z-A_z)E[l-1,m,\tilde{l},\tilde{m},t,u,v] \nonumber \\&\quad +\,2(t+1)(P_x-A_x)G_z^A[l-1,m,\tilde{l},\tilde{m},t+1,u,v] \nonumber \\&\quad +\,2(u+1)(P_y-A_y)G_z^A[l-1,m,\tilde{l},\tilde{m},t,u+1,v] \nonumber \\&\quad +\,2(v+1)(P_z-A_z)G_z^A[l-1,m,\tilde{l},\tilde{m},t,u,v+1] \nonumber \\&\quad -\,2(v+1)E[l-1,m,\tilde{l},\tilde{m},t,u,v+1] \nonumber \\&\quad +\,(t+2)(t+1)G_z^A[l-1,m,\tilde{l},\tilde{m},t+2,u,v] \nonumber \\&\quad +\,(u+2)(u+1)G_z^A[l-1,m,\tilde{l},\tilde{m},t,u+2,v] \nonumber \\&\quad +\,(v+2)(v+1)G_z^A[l-1,m,\tilde{l},\tilde{m},t,u,v+2] \bigg \} \end{aligned}$$

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