Molecular Integrals

Contains the information about how the integrals are calculated. In the equations in this section the following definitions is used.

\[p=a+b\]
\[\mu=\frac{ab}{a+b}\]
\[P_{x}=\frac{aA_{x}+bB_{x}}{p}\]
\[X_{AB}=A_{x}-B_{x}\]

Here a and b are Gaussian exponent factors. Ax and Bx are the position of the Gaussians in one dimension. Further the basisset functions is of Gaussian kind and given as:

\[\phi_{A}\left(r\right)=N\left(x-A_{x}\right)^{l}\left(y-A_{y}\right)^{m}\left(z-A_{z}\right)^{n}\exp\left(-\zeta\left(\vec{r}-\vec{A}\right)^{2}\right)\]

with a normalization constant given as:

\[N=\left(\frac{2\alpha}{\pi}\right)^{3/4}\left[\frac{\left(8\alpha\right)^{l+m+n}l!m!n!}{\left(2l\right)!\left(2m\right)!\left(2n\right)!}\right]\]

Boys Function

The Boys function is given as:

\[F_{n}\left(x\right)=\int_{0}^{1}\exp\left(-xt^{2}\right)t^{2n}dt\]

FUNCTION:

  • MIcython.boys(m,T)
  • return value

Input:

  • m, subscript of the Boys function
  • T, argument of the Boys function

Output:

  • value, value corrosponding to given m and T

References:

  • Trygve Helgaker, Poul Jorgensen and Jeppe Olsen, Molecular Electronic-Structure Theory

Expansion coefficients

The expansion coefficient is found by the following recurrence relation:

\[E_{t}^{i,j}=0,\,\,\,\,t<0\,\,\mathrm{or}\,\,t>i+j\]
\[E_{t}^{i+1,j}=\frac{1}{2p}E_{t-1}^{i,j}+X_{\mathrm{PA}}E_{t}^{i,j}+\left(t+1\right)E_{t+1}^{i,j}\]
\[E_{t}^{i,j+1}=\frac{1}{2p}E_{t-1}^{i,j}+X_{\mathrm{PB}}E_{t}^{i,j}+\left(t+1\right)E_{t+1}^{i,j}\]

With the boundary condition that:

\[E_{0}^{0,0}=\exp\left(-pX_{\mathrm{AB}}^{2}\right)\]

FUNCTION:

  • MolecularIntegrals.E(i,j,t,Qx,a,b,XPA,XPB,XAB)
  • return val

Input:

  • i, input values
  • j, input values
  • t, input values
  • Qx, input values
  • a, input values
  • b, input values
  • XPA, input values
  • XPB, input values
  • XAB, input values

Output:

  • val, value corrosponding to the given input

References:

Hermite coulomb integral

The hermite coulomb integrals is given as the following recurrence relations:

\[R_{t+1,u,v}^{n}\left(p,R_{\mathrm{PC}}\right)=tR_{t-1,u,v}^{n+1}\left(p,R_{\mathrm{PC}}\right)+X_{\mathrm{PC}}R_{t,u,v}^{n+1}\left(p,R_{\mathrm{PC}}\right)\]
\[R_{t,u+1,v}^{n}\left(p,R_{\mathrm{PC}}\right)=uR_{t,u-1,v}^{n+1}\left(p,R_{\mathrm{PC}}\right)+Y_{\mathrm{PC}}R_{t,u,v}^{n+1}\left(p,R_{\mathrm{PC}}\right)\]
\[R_{t,u,v+1}^{n}\left(p,R_{\mathrm{PC}}\right)=vR_{t,u,v-1}^{n+1}\left(p,R_{\mathrm{PC}}\right)+Z_{\mathrm{PC}}R_{t,u,v}^{n+1}\left(p,R_{\mathrm{PC}}\right)\]

With the boundary condition:

\[R_{0,0,0}^{n}\left(p,R_{\mathrm{PC}}\right)=\left(-2p\right)^{n}F_{n}\left(pR_{\mathrm{PC}}^{2}\right)\]

FUNCTION:

  • MIcython.R2(t,u,v,n,p,PCx,PCy,PCz,RPC)
  • return val

Input:

  • t, input value
  • u, input value
  • v, input value
  • n, input value
  • p, input value
  • PCx, input value
  • PCy, input value
  • PCz, input value
  • RPC, input value

Output:

  • val, value corrosponding to the given input

References:

Overlap

The overlap integrals are solved by the following recurrence relation:

\[S_{i+1,j}=X_{PA}S_{ij}+\frac{1}{2p}\left(iS_{i-1,j}+jS_{i,j-1}\right)\]
\[S_{i,j+1}=X_{PB}S_{ij}+\frac{1}{2p}\left(iS_{i-1,j}+jS_{i,j-1}\right)\]

With the boundary condition that:

\[S_{00}=\sqrt{\frac{\pi}{p}}\exp\left(-\mu X_{AB}^{2}\right)\]

FUNCTION:

  • MolecularIntegrals.Overlap(a, b, la, lb, Ax, Bx)
  • return Sij

Input:

  • a, Gaussian exponent factor
  • b, Gaussian exponent factor
  • la, angular momentum quantum number
  • lb, angular momentum quantum number
  • Ax, position along one axis
  • Bx, position along one axis

Output:

  • Sij, non-normalized overlap element in one dimension

References:

  • Trygve Helgaker, Poul Jorgensen and Jeppe Olsen, Molecular Electronic-Structure Theory

Kinetic energy

The kinetic energy integrals are solved by the following recurrence relation:

\[T_{i+1,j}=X_{\mathrm{PA}}T_{i,j}+\frac{1}{2p}\left(iT_{i-1,j}+jT_{i,j-1}\right)+\frac{b}{p}\left(2aS_{i+1,j}-iS_{i-1,j}\right)\]
\[T_{i,j+1}=X_{\mathrm{PB}}T_{i,j}+\frac{1}{2p}\left(iT_{i-1,j}+jT_{i,j-1}\right)+\frac{a}{p}\left(2bS_{i,j+1}-iS_{i,j-1}\right)\]

With the boundary condition that:

\[T_{00}=\left[a-2a^{2}\left(X_{\mathrm{PA}}^{2}+\frac{1}{2p}\right)\right]S_{00}\]

FUNCTION:

  • Kin(a, b, Ax, Ay, Az, Bx, By, Bz, la, lb, ma, mb, na, nb, N1, N2, c1, c2)
  • return Tij, Sij

Input:

  • a, Gaussian exponent factor
  • b, Gaussian exponent factor
  • Ax, position along the x-axis
  • Bx, position along the x-axis
  • Ay, position along the y-axis
  • By, position along the y-axis
  • Az, position along the z-axis
  • Bz, position along the z-axis
  • la, angular momentum quantum number
  • lb, angular momentum quantum number
  • ma, angular momentum quantum number
  • mb, angular momentum quantum number
  • na, angular momentum quantum number
  • nb, angular momentum quantum number
  • N1, normalization constant
  • N2, normalization constant
  • c1, Gaussian prefactor
  • c2, Gaussian prefactor

Output:

  • Tij, normalized kinetic energy matrix element
  • Sij, normalized overlap matrix element

References:

  • Trygve Helgaker, Poul Jorgensen and Jeppe Olsen, Molecular Electronic-Structure Theory

Electron-nuclear attraction

The electron-nuclear interaction integral is given as:

\[V_{ijklmn}^{000}=\frac{2\pi}{p}\sum_{t}^{i+j}E_{t}^{ij}\sum_{u}^{k+l}E_{u}^{kl}\sum_{v}^{m+n}E_{v}^{mn}R_{tuv}\]

FUNCTION:

  • MolecularIntegrals.elnuc(P, p, l1, l2, m1, m2, n1, n2, N1, N2, c1, c2, Zc, Ex, Ey, Ez, R1)
  • return Vij

Input:

  • P, Gaussian product
  • p, exponent from Guassian product
  • l1, angular momentum quantum number
  • l2, angular momentum quantum number
  • m1, angular momentum quantum number
  • n1, angular momentum quantum number
  • n2, angular momentum quantum number
  • N1, normalization constant
  • N2, normalization constant
  • c1, Gaussian prefactor
  • c2, Gaussian prefactor
  • Zc, Nuclear charge
  • Ex, expansion coefficients
  • Ey, expansion coefficients
  • Ez, expansion coefficients
  • R1, hermite coulomb integrals

Output:

  • Vij, normalized electron-nuclei attraction matrix element

References:

Electron-nuclear field

The electron-nuclear interaction integral is given as:

\[V_{ijklmn}^{efg}=\left(-1\right)^{e+f+g}\frac{2\pi}{p}\sum_{t}^{i+j}E_{t}^{ij}\sum_{u}^{k+l}E_{u}^{kl}\sum_{v}^{m+n}E_{v}^{mn}R_{t+e,u+f,v+g}\]

Here e, f and g detones the order of derivate with respect to x, y and z

FUNCTION:

  • MolecularIntegrals.electricfield(p, Ex, Ey, Ez, Zc, l1, l2, m1, m2, n1, n2, N1, N2, c1, c2, derivative, R1)
  • return VijA

Input:

  • p, Gaussian exponent form Gaussian product
  • Ex, expansion coefficient
  • Ey, expansion coefficient
  • Ez, expansion coefficient
  • l1, angular momentum quantum number
  • l2, angular momentum quantum number
  • m1, angular momentum quantum number
  • n1, angular momentum quantum number
  • n2, angular momentum quantum number
  • N1, normalization constant
  • N2, normalization constant
  • c1, Gaussian prefactor
  • c2, Gaussian prefactor
  • derivative, axis of derivative (dx,dy or dz)
  • R1, hermite coulomb integral

Output:

  • VijA, normalized electron-nuclei field of nuclei A matrix element

References:

  • Trygve Helgaker, Poul Jorgensen and Jeppe Olsen, Molecular Electronic-Structure Theory

Electron-electron repulsion

The electron-electron repulsion integral is calculated as:

\[g_{abcd}=\sum_{t}^{l1+l2}E_{t}^{ab}\sum_{u}^{m1+m2}E_{u}^{ab}\sum_{v}^{n1+n2}E_{v}^{ab}\sum_{\tau}^{l3+l4}E_{\tau}^{cd}\sum_{\nu}^{m3+m4}E_{\nu}^{cd}\sum_{\phi}^{n3+n4}E_{\phi}^{cd}\left(-1\right)^{\tau+\nu+\phi}\frac{2\pi^{5/2}}{pq\sqrt{p+q}}R_{t+\tau,u+\nu,v+\phi}\left(\alpha,R_{\mathrm{PQ}}\right)\]

FUNCTION:

  • MIcython.elelrep(p, q, l1, l2, l3, l4, m1, m2, m3, m4, n1, n2, n3, n4, N1, N2, N3, N4, c1, c2, c3, c4, E1, E2, E3, E4, E5, E6, Rpre)
  • return Veeijkl

Input:

  • p, Gaussian exponent factor from Gaussian product
  • q, Gaussian exponent factor from Gaussian product
  • l1, angular momentum quantum number
  • l2, angular momentum quantum number
  • l3, angular momentum quantum number
  • l4, angular momentum quantum number
  • m1, angular momentum quantum number
  • m2, angular momentum quantum number
  • m3, angular momentum quantum number
  • m4, angular momentum quantum number
  • n1, angular momentum quantum number
  • n2, angular momentum quantum number
  • n3, angular momentum quantum number
  • n4, angular momentum quantum number
  • N1, normalization constant
  • N2, normalization constant
  • N3, normalization constant
  • N4, normalization constant
  • c1, Gaussian prefactor
  • c2, Gaussian prefactor
  • c3, Gaussian prefactor
  • c4, Gaussian prefactor
  • E1, expansion coefficient
  • E2, expansion coefficient
  • E3, expansion coefficient
  • E4, expansion coefficient
  • E5, expansion coefficient
  • E6, expansion coefficient
  • Rpre, hermite coulomb integral

Output:

  • Veeijkl, normalized electron-electron repulsion matrix element

References:

Nuclear-nuclear repulsion

The nucleus-nucleus repulsion term is calculated with classical nuclie as follows:

\[V_{NN}=\sum_{A}\sum_{B<A}\frac{Z_{A}Z_{B}}{r_{AB}}\]

FUNCTION:

  • MolecularIntegrals.nucrep(input)
  • return Vnn

Input:

  • input, inputfile object

Output:

  • Vnn, nuclear repulsion energy

References:

  • None

Nuclear-nuclear field

The nucleus-nucleus field term is calculated with classical nuclie as follows:

\[\frac{\partial V_{NN}}{\partial X_{A}}=-Z_{A}\sum_{B\neq A}\frac{Z_{B}\left(X_{B}-X_{A}\right)}{r_{AB}^{3}}\]

FUNCTION:

  • MolecularIntegrals.nucdiff(input, atomidx, direction)
  • return Vnn

Input:

  • input, inputfile object
  • atomidx, atom which is differentiated with respect to
  • direction, axis of differentiation (1 = dx, 2 = dy, 3 = dz)

Output:

  • Vnn, nucleus-nucleus field

References:

  • None

Dipole moment integral

The dipole moment integral is calculated by using the following relations:

\[S_{i+1,j}^{e}=X_{\mathrm{PA}}S_{i,j}^{e}+\frac{1}{2p}\left(iS_{i-1,j}^{e}+jS_{i,j-1}^{e}+eS_{ij}^{e-1}\right)\]
\[S_{i,j+1}^{e}=X_{\mathrm{PB}}S_{i,j}^{e}+\frac{1}{2p}\left(iS_{i-1,j}^{e}+jS_{i,j-1}^{e}+eS_{ij}^{e-1}\right)\]
\[S_{i,j}^{e+1}=X_{\mathrm{PC}}S_{i,j}^{e}+\frac{1}{2p}\left(iS_{i-1,j}^{e}+jS_{i,j-1}^{e}+eS_{ij}^{e-1}\right)\]

Here e is the order of multipole moment, e=1 is dipole moment.

FUNCTION:

  • MolecularIntegrals.u_ObaraSaika(a1, a2, Ax, Ay, Az, Bx, By, Bz, la, lb, ma, mb, na, nb, N1, N2, c1, c2, input)
  • return muxij, muyij, muzij

Input:

  • a1, Gaussian exponent factor
  • a2, Gaussian exponent factor
  • Ax, position along x axis
  • Ay, position along y axis
  • Az, position along z axis
  • Bx, position along x axis
  • By, position along y axis
  • Bz, position along z axis
  • la, angular momentum quantum number
  • lb, angular momentum quantum number
  • ma, angular momentum quantum number
  • mb, angular momentum quantum number
  • na, angular momentum quantum number
  • nb, angular momentum quantum number
  • N1, normalization constant
  • N2, normalization constant
  • c1, Gaussian prefactor
  • c2, Gaussian prefactor
  • input, inputfile object

Output:

  • muxij, dipolemoment integral matrix element for x axis
  • muyij, dipolemoment integral matrix element for y axis
  • muzij, dipolemoment integral matrix element for z axis

References:

  • Trygve Helgaker, Poul Jorgensen and Jeppe Olsen, Molecular Electronic-Structure Theory