Class ECPIntegral

Class Documentation

class ECPIntegral

Calculates ECP integrals.

Given an ECP basis, and orbital bases, this will calculate the ECP integrals over all ECP centers.

REFERENCES: (Shaw2017) R. A. Shaw, J. G. Hill, J. Chem. Phys. 147 (2017), 074108 (Flores06) R. Flores-Moreno et al., J. Comput. Chem. 27 (2006), 1009 (MM81) L. E. McMurchie and E. R. Davidson, J. Comp. Phys. 44 (1981), 289 - 301

Public Functions

void makeC(FiveIndex<double> &C, int L, const double *A) const

Constructs the coefficients in the binomial expansion (see REF. Shaw2017)

Parameters

  • C – - reference to a FiveIndex array to store the results in

  • L – - maximum angular momentum to go up to in expansion

  • A – - xyz coordinates for the center to calculate over

ECPIntegral(int maxLB, int maxLU, int deriv = 0)

Creates an ECP integrator, initialising the radial and angular parts for subsequent calculations.

Parameters

  • maxLB – - the maximum angular momentum in the orbital basis

  • maxLU – - the maximum angular momentum in the ECP basis

  • deriv – - the maximum order of derivative to be calculated (TODO: derivs currently being implemented)

void type1(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, const ShellPairData &data, const FiveIndex<double> &CA, const FiveIndex<double> &CB, const RadialIntegral::Parameters &parameters, TwoIndex<double> &values) const

Calculates the type 1 integrals for the given ECP center over the given shell pair, using quadrature

Parameters

  • U – - reference to ECP

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • data – - wrapper for data about shell pair

  • CA – - binomial expansion coefficients for shellA, made with makeC

  • CB – - binomial expansion coefficients for shellB, made with makeC

  • parameters – - pre-calculated parameters for the radial integral

  • values – - array in which results are returned

void type2(int l, const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, const ShellPairData &data, const FiveIndex<double> &CA, const FiveIndex<double> &CB, const RadialIntegral::Parameters &parameters, ThreeIndex<double> &values) const

Calculates the type 2 integrals for the given ECP center over the given shell pair

Parameters

  • l – - angular momentum shell of ECP to calculate over

  • U – - reference to ECP

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • data – - wrapper for data about shell pair

  • CA – - binomial expansion coefficients for shellA, made with makeC

  • CB – - binomial expansion coefficients for shellB, made with makeC

  • parameters – - pre-calculated parameters for the radial integral

  • values – - array in which results are returned

void estimate_type2(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, const ShellPairData &data, double *results) const
void compute_shell_pair(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, TwoIndex<double> &values, int shiftA = 0, int shiftB = 0) const

Computes the overall ECP integrals over the given ECP center and shell pair. This is the lower level API, where you want finer control over the calculation. Results are returned with rows corresponding to shellA and cols to shellB, with the Cartesian functions in alpha order e.g. {xxx, xxy, xxz, xyy, xyz, xzz, yyy, yyz, yzz, zzz} l = 3

Parameters

  • U – - reference to the ECP to calculate the integral over

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • values – - reference to TwoIndex array where the results will be stored

void compute_shell_pair_derivative(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, std::array<TwoIndex<double>, 9> &results) const

Computes the overall ECP integral first derivatives over the given ECP center, C, and shell pair (A | B) The results are placed in order [Ax, Ay, Az, Bx, By, Bz, Cx, Cy, Cz] and are calculated so that each component can always be added to the relevant total derivative. E.g. if A = B, then the contribution to the total derivative for that coordinate on the x axis will be Ax + Bx. The order for each derivative matrices matches that specified in compute_shell_pair

Parameters

  • U – - reference to the ECP

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • results – - reference to array of 9 TwoIndex arrays where the results will be stored

void compute_shell_pair_second_derivative(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, std::array<TwoIndex<double>, 45> &results) const

Computes the overall ECP integral second derivatives over the given ECP center, C, and shell pair (A | B) The results are placed in order [AA, AB, AC, BB, BC, CC] with components [xx, xy, xz, yy, yz, zz] for AA, BB, and CC, and [xx, xy, xz, yx, yy, yz, zx, zy, zz] for AB, AC, and BC. As for the first derivatives, the components are calculated such that they can usually be added to the relevant total derivative. However, this is more complicated than for first derivatives, especially in the instance where A=B. It’s recommended to look at the compute_second_derivatives interface in api.cpp for how to handle this. The order for each derivative matrices matches that specified in compute_shell_pair

Parameters

  • U – - reference to the ECP

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • results – - reference to array of 45 TwoIndex arrays where the results will be stored

void left_shell_derivative(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, std::array<TwoIndex<double>, 3> &results) const

Worker function to calculate the derivative of the integral <A | C | B> with respect to A. This is given as <d_q A(l_q) | C | B> = l_q*<A(l_q-1) | C | B> - 2*mu*<A(l_q + 1) | C | B> where l_q is the angular momentum component of A in the q coordinate, and mu is the exponent of A.

Parameters

  • U – - reference to the ECP

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • results – - reference to array of 3 TwoIndex arrays for the [x, y, z] derivatives

void left_shell_second_derivative(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, std::array<TwoIndex<double>, 6> &results) const

Worker function to calculate the second derivatives of the integral <A | C | B> with respect to AA. This is given as <d_p d_q A(l_p, l_q) | C | B> = l_p*l_q*<A(l_p-1, l_q-1) | C | B> - 2*mu*l_p*<A(l_p-1, l_q+1) | C | B>

  • 2*mu*l_q*<A(l_p+1, l_q-1) | C | B> + 4*mu^2*<A(l_p+1, l_q+1) | C | B >

Parameters

  • U – - reference to the ECP

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • results – - reference to array of 6 TwoIndex arrays for the [xx, xy, xz, yy, yz, zz] derivatives

void mixed_second_derivative(const ECP &U, const GaussianShell &shellA, const GaussianShell &shellB, std::array<TwoIndex<double>, 9> &results) const

Worker function to calculate the second derivatives of the integral <A | C | B> with respect to AB. This is given as <d_p A(l_p) | C | d_q B(l_q)> = l_p*l_q*<A(l_p-1) | C | B(l_q-1)> - 2*mu_B*l_p*<A(l_p-1) | C | B(l_q+1)>

  • 2*mu_A*l_q*<A(l_p+1) | C | B(l_q-1)> + 4*mu_A*mu_B*<A(l_p+1) | C | B(l_q+1) >

Parameters

  • U – - reference to the ECP

  • shellA – - the first basis shell (rows in values)

  • shellB – - the second basis shell (cols in values)

  • results – - reference to array of 9 TwoIndex arrays for the [xx, xy, xz, yx, yy, yz, zx, zy, zz] derivatives

Public Members

int skipped
int zero
int nonzero