Integrals

openLOWDIN computes analitically one-body and two-body integrals based on atomic orbitals expressed as a sum of contracted GTFs under a cartesian coordinate representation (see Basis)

\begin{equation} \phi_i^{\alpha}(\mathbf{r}_i) = \sum_{\mu}^{N_{bas}^{\alpha}} C_{\mu}^{\alpha} \varphi_{\mu}^{\alpha}(\mathbf{r}_i;\mathbf{R}_{\mu}) \end{equation}

In matrix form, the following integrals are implemented in openLOWDIN

One-body integrals

All computed under an Obara-Saika scheme for any angular momentum in the basis set

Overlap

\begin{equation} S_{\mu \nu}^\alpha =\int d r_i \varphi_\mu^\alpha(i) \varphi_\nu^\alpha(i) \end{equation}

Kinetic

\begin{equation} T_{\mu \nu}^\alpha =\int d r_i \varphi_\mu^\alpha(i) \nabla_i^2 \varphi_\nu^\alpha(i) \end{equation}

Point charge potential

\begin{equation} \mathbb{PC}_{\mu \nu}^\alpha = \sum_{I} \int d r_i \varphi_\mu^\alpha(i) \frac{1}{r_i - R_I} \varphi_\nu^\alpha(i) \end{equation}

Moment integrals

\begin{align} \mathbb{UA}_{\mu \nu}^\alpha = \int d r_i \varphi_\mu^\alpha(i) r_{i,\mathbb{A}} \varphi_\nu^\alpha(i) \quad \mathbb{A} = { x,y,z } \\ \mathbb{UAB}_{\mu \nu}^\alpha = \int d r_i \varphi_\mu^\alpha(i) r_{i,\mathbb{A}} r_{i,\mathbb{B}} \varphi_\nu^\alpha(i) \quad \mathbb{A,B} = { x,y,z } \end{align}

Harmonic integrals

\begin{align} \mathbb{HA}_{\mu \nu}^\alpha = \int d r_i \varphi_\mu^\alpha(i) r^2_{i} \varphi_\nu^\alpha(i) \end{align}

Three-center integrals

\begin{align} \mathbb{3C}_{\mu \nu}^\alpha = \sum_{\sigma} \int d r_i \varphi_\mu^\alpha(i) \varphi_{\sigma}(i) \varphi_\nu^\alpha(i) \end{align}

Two-body integrals

These integrals are computed with LIBINT library https://github.com/evaleev/libint

Four-center intraspecies coulomb potential

\begin{align} \mathbb{4C}_{\mu \nu \sigma \lambda }^{\alpha} &= \langle \mu^\alpha \nu^\alpha | \sigma^\alpha\lambda^\alpha \rangle \\ &= \int \int d r_i d r_j \varphi_\mu^\alpha(i) \varphi_\nu^\alpha(j) \frac{1}{r_i - r_j} \varphi_\sigma^\alpha(i) \varphi_\lambda^\alpha(j) \end{align}

Four-center interspecies coulomb potential

\begin{align} \mathbb{4C}_{\mu \nu \sigma \lambda }^{\alpha\beta} &= \langle \mu^\alpha \nu^\beta | \sigma^\alpha\lambda^\beta \rangle \\ & = \int \int d r_i d r_j \varphi_\mu^\alpha(i) \varphi_\nu^\beta(j) \frac{1}{r_i - r_j} \varphi_\sigma^\alpha(i) \varphi_\lambda^\beta(j) \end{align}

Five-center, intra- and interspecies

\begin{align} \mathbb{5C}_{\mu \nu \sigma \lambda }^{\alpha,\beta} &= \langle \mu^\alpha \nu^\beta | V_2^{\alpha\beta}(\mathbf{r}^{\alpha}_i,\mathbf{r}^{\beta}_j) | \sigma^\alpha\lambda^\beta \rangle \\ &= \sum_\tau C_\tau^{\alpha\beta} \int \int d r_i d r_j \varphi_\mu^\alpha(i) \varphi_\nu^\beta(j) \ \text{exp}[ -a_\tau^{\alpha\beta} ({r}_i - {r}_{j})^2 ] \varphi_\sigma^\alpha(i) \varphi_\lambda^\beta(j) \end{align}

Input options

• tv= [float] deprecated Default 1.0E-6

• integralThreshold= [float] threshold to store integrals in disk above the given value. Default 1.0E-10

• integralStackSize= [integer] write and load integrals temporary files by stacks of this values. Default 30000

• integralStorage= [character] select storage scheme for two-particles integrals

  "DISK"	Storage all non-zero integrals in disk at $SCRATCH folder, after four-index permutational symmetries. Default
  "MEMORY"	Allocate a four dimensional array in RAM memory
  "DIRECT"	Compute integrals on-the-fly (only for the SCF step)

• integralScheme= [character] select two-particles library. Default "LIBINT"

• schwarzInequality= [logical] perfoms Schwarz inequality to skip blocks of small integrals. Deprecated, now it’s used by default within libint interface. Default .false.