mimic_tensors

This module provides procedures for computing interactions tensors in packed form (one-dimensional arrays) and includes several operations involving tensors.

Uses

- mimic_precision

Variables

Type	Visibility	Attributes		Name		Initial
integer,	private,	parameter	::	MAX_ORDER	=	42	maximum order of derivative
real(kind=dp),	private,	dimension(0:MAX_ORDER,0:MAX_ORDER,1:2*MAX_ORDER+1), save	::	tensor_coefficient	=	huge(0.0_dp)	needed for open-ended computation of derivatives of $\frac{1}{\|\mathbf{r}_{a}-\mathbf{r}_{b}\|}$
real(kind=dp),	public,	dimension(0:MAX_ORDER), save	::	factorial	=	huge(0.0_dp)	factorials computed and stored during initialization
real(kind=dp),	public,	dimension(-1:MAX_ORDER), save	::	double_factorial	=	huge(0.0_dp)	double factorials computed and stored during initialization
real(kind=dp),	public,	dimension(0:3*MAX_ORDER,0:MAX_ORDER), save	::	binomial	=	huge(0.0_dp)	binomial coefficients computed and stores during initialization
real(kind=dp),	public,	dimension(0:MAX_ORDER,0:MAX_ORDER,0:MAX_ORDER), save	::	trinomial	=	huge(0.0_dp)	trinomial coefficients computed and stores during initialization

Functions

public pure function tensor_element(i, j, k, r)

compute element of multipole interaction tensor (C. E. Dykstra, J. Comput. Chem., 9 (1988), 476) which in multi-index notation this is $\mathbf{T}^{(i,j,k)} = \frac{\mathrm{d}^{i}\mathrm{d}^{j}\mathrm{d}^{k}}{\mathrm{d}x_{a}^{i}\mathrm{d}y_{a}^{j}\mathrm{d}z_{a}^{k}} \frac{1}{|\mathbf{r}_{a} - \mathbf{r}_{b}|}$

Arguments

Type	Intent	Attributes		Name
integer,	intent(in)		::	i	order of the derivative wrt $x$ -component of $\mathbf{r}_{a}$
integer,	intent(in)		::	j	order of the derivative wrt $y$ -component of $\mathbf{r}_{a}$
integer,	intent(in)		::	k	order of the derivative wrt $z$ -component of $\mathbf{r}_{a}$
real(kind=dp),	intent(in),	dimension(:)	::	r	coordinate vector $\mathbf{r} = \mathbf{r}_{a}$ - $\mathbf{r}_{b}$

Return Value real(kind=dp)

public pure function tensor_trace(A, ax, ay, az, order)

compute trace of a packed tensor

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in),		dimension(:)	::	A
integer,	intent(in)			::	ax
integer,	intent(in)			::	ay
integer,	intent(in)			::	az
integer,	intent(in),	optional		::	order

Return Value real(kind=dp)

public pure function convert_index(ax, ay, az)

convert from multi-index to packed tensor index

Arguments

Type	Intent		Name
integer,	intent(in)	::	ax	first ( $x$ ) component of multi-index
integer,	intent(in)	::	ay	second ( $y$ ) component of multi-index
integer,	intent(in)	::	az	third ( $z$ ) component of multi-index

Return Value integer

public pure function tensor_rank(A)

compute rank of tensor

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in),		dimension(:)	::	A

Return Value integer

public pure function tensor_size(rank)

compute size of tensor of given rank

Arguments

Type	Intent	Optional	Attributes		Name
integer,	intent(in)			::	rank

Return Value integer

public pure function polytensor_size(max_rank)

compute size of first degree polytensor with tensors up to maxmimum rank

Arguments

Type	Intent	Optional	Attributes		Name
integer,	intent(in)			::	max_rank

Return Value integer

Subroutines

public subroutine initialize_tensors()

initialize tensor module

Arguments

None

public subroutine terminate_tensors()

Arguments

None

private subroutine compute_tensor_coefficients()

compute tensor coefficients needed to compute open-ended derivatives of $\frac{1}{|\mathbf{r}_{a}-\mathbf{r}_{b}|}$ (from C. E. Dykstra, J. Comput. Chem., 9 (1988), 476, where C^(n)_ij are stored as tensor_coefficient(j,i,n))

Arguments

None

private subroutine compute_factorials()

compute and store factorials upto 12!

Arguments

None

private subroutine compute_double_factorials()

compute and store double factorials upto 12!!

Arguments

None

private subroutine compute_binomial_coefficients()

Calculate binomial coefficients using recursive algorithm by Sten Rettrup and Ruben Pauncz, Int. J. Quantum Chem., 60: 91–98 (1996) DOI: 10.1002/(SICI)1097-461X(1996)60:1<91::AID-QUA10>3.0.CO;2-A

Arguments

None

private subroutine compute_trinomial_coefficients()

compute trinomial coefficients ( $\frac{(i+j+k)!}{i!+j!+k!}$ )

Arguments

None

public pure subroutine contract_tensors(A, B, C)

contract tensors A and B according to rank of C and return result in C

Arguments

Type	Intent	Attributes		Name
real(kind=dp),	intent(in),	dimension(:)	::	A
real(kind=dp),	intent(in),	dimension(:)	::	B
real(kind=dp),	intent(inout),	dimension(:)	::	C

public pure subroutine interaction_tensor(r_ab, T, da, db)

Compute multipole interaction tensor containing derivatives of $\frac{1}{r_{ab}} = \frac{1}{|\mathbf{r}_{b}-\mathbf{r}_{a}|}$ , i.e. the da'th derivative wrt $\mathbf{r}_{a}$ and db'th derivative wrt $\mathbf{r}_{b}$ . Tensor is returned in a one-dimensional array with elements in anticanonical ordering, e.g. xx, xy, xz, yy, yz, zz.

Arguments

Type	Intent	Attributes		Name
real(kind=dp),	intent(in),	dimension(:)	::	r_ab	coordinate $\mathbf{r}_{ab} = \mathbf{r}_{b}-\mathbf{r}_{a}$
real(kind=dp),	intent(out),	dimension(:)	::	T	the interaction tensor
integer,	intent(in)		::	da	order of derivative wrt $\mathbf{r}_{a}$
integer,	intent(in)		::	db	order of derivative wrt $\mathbf{r}_{b}$

public pure subroutine folded_interaction_tensor(r_ab, T, da, db)

Compute folded multipole interaction tensor containing derivatives of $\frac{1}{r_{ab}} = \frac{1}{|\mathbf{r}_{b}-\mathbf{r}_{a}|}$ , i.e. the da'th derivative wrt $\mathbf{r}_{a}$ and db'th derivative wrt $\mathbf{r}_{b}$ . Tensor is returned in a one-dimensional array with elements in anticanonical ordering, e.g. xx, xy, xz, yy, yz, zz.

Arguments

Type	Intent	Attributes		Name
real(kind=dp),	intent(in),	dimension(:)	::	r_ab	coordinate $\mathbf{r}_{ab} = \mathbf{r}_{b}-\mathbf{r}_{a}$
real(kind=dp),	intent(out),	dimension(:)	::	T	the interaction tensor
integer,	intent(in)		::	da	order of derivative wrt $\mathbf{r}_{a}$
integer,	intent(in)		::	db	order of derivative wrt $\mathbf{r}_{b}$

public pure subroutine detrace_tensor(A)

convert tensor to traceless tensor

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(inout),		dimension(:)	::	A

mimic_tensors Module

Contents

Variables

Functions

Subroutines

Uses

Contents

Variables

Functions

Subroutines

Variables

Functions

public pure function tensor_element(i, j, k, r)

Arguments

Return Value real(kind=dp)

public pure function tensor_trace(A, ax, ay, az, order)

Arguments

Return Value real(kind=dp)

public pure function convert_index(ax, ay, az)

Arguments

Return Value integer

public pure function tensor_rank(A)

Arguments

Return Value integer

public pure function tensor_size(rank)

Arguments

Return Value integer

public pure function polytensor_size(max_rank)

Arguments

Return Value integer

Subroutines

public subroutine initialize_tensors()

Arguments

public subroutine terminate_tensors()

Arguments

private subroutine compute_tensor_coefficients()

Arguments

private subroutine compute_factorials()

Arguments

private subroutine compute_double_factorials()

Arguments

private subroutine compute_binomial_coefficients()

Arguments

private subroutine compute_trinomial_coefficients()

Arguments

public pure subroutine contract_tensors(A, B, C)

Arguments

public pure subroutine interaction_tensor(r_ab, T, da, db)

Arguments

public pure subroutine folded_interaction_tensor(r_ab, T, da, db)

Arguments

public pure subroutine detrace_tensor(A)

Arguments