Package ffx.numerics.multipole

Class MultipoleTensor

java.lang.Object

ffx.numerics.multipole.MultipoleTensor

Direct Known Subclasses:

CoulombTensorGlobal, CoulombTensorQI

public abstract class MultipoleTensor
extends Object

The abstract MultipoleTensor is extended by classes that compute derivatives of 1/|r| via recursion to arbitrary order using Cartesian multipoles in either a global frame or a quasi-internal frame.
This class serves as the abstract parent to both and defines all shared logic. Non-abstract methods are declared final to disallow unnecessary overrides.

Since:

1.0

Author:

Michael J. Schnieders

Field Summary

Fields

Modifier and Type	Field	Description
protected final CoordinateSystem	coordinates	The coordinate system in use (global or QI).
protected final double[]	coulombSource	These are the "source" terms for the recursion for the Coulomb operator (1/R).
protected final int	il	Order plus one.
protected final int	im	im = (Order plus one)^2.
protected final int	in	in = (Order plus one)^3.
protected final int	o1	Order plus 1.
protected Operator	operator	The OPERATOR in use.
protected final int	order	Order of the tensor recursion (5th is needed for AMOEBA forces).
protected double	R	Separation distance.
protected double	r2	Separation distance squared.
protected final int	size	Size = (order + 1) * (order + 2) * (order + 3) / 6;
protected final int	t000	No derivatives.
protected final double[]	T000	Store the auxiliary tensor memory to avoid memory consumption.
protected final int	t001	First derivative with respect to z.
protected final int	t002	Second derivative with respect to z.
protected final int	t003	Third derivative with respect to z.
protected final int	t004	Fourth derivative with respect to z.
protected final int	t005	Fifth derivative with respect to z.
protected final int	t006	Sixth derivative with respect to z.
protected final int	t010	First derivative with respect to y.
protected final int	t011	Derivatives with respect to y and z.
protected final int	t012	Derivatives with respect to y and z.
protected final int	t013	Derivatives with respect to y and z.
protected final int	t014	Derivatives with respect to y and z.
protected final int	t015	Derivatives with respect to y and z.
protected final int	t020	Second derivative with respect to y.
protected final int	t021	Derivatives with respect to y and z.
protected final int	t022	Derivatives with respect to y and z.
protected final int	t023	Derivatives with respect to y and z.
protected final int	t024	Derivatives with respect to y and z.
protected final int	t030	Third derivative with respect to y.
protected final int	t031	Derivatives with respect to y and z.
protected final int	t032	Derivatives with respect to y and z.
protected final int	t033	Derivatives with respect to y and z.
protected final int	t040	Fourth derivative with respect to y.
protected final int	t041	Derivatives with respect to y and z.
protected final int	t042	Derivatives with respect to y and z.
protected final int	t050	Fifth derivative with respect to y.
protected final int	t051	Derivatives with respect to y and z.
protected final int	t060	Sixth derivative with respect to y.
protected final int	t100	First derivative with respect to x.
protected final int	t101	Derivatives with respect to x and z.
protected final int	t102	Derivatives with respect to x and z.
protected final int	t103	Derivatives with respect to x and z.
protected final int	t104	Derivatives with respect to x and z.
protected final int	t105	Derivatives with respect to x and z.
protected final int	t110	Derivatives with respect to x and y.
protected final int	t111	Derivatives with respect to x, y and z.
protected final int	t112	Derivatives with respect to x, y and z.
protected final int	t113	Derivatives with respect to x, y and z.
protected final int	t114	Derivatives with respect to x, y and z.
protected final int	t120	Derivatives with respect to x and y.
protected final int	t121	Derivatives with respect to x, y and z.
protected final int	t122	Derivatives with respect to x, y and z.
protected final int	t123	Derivatives with respect to x, y and z.
protected final int	t130	Derivatives with respect to x and y.
protected final int	t131	Derivatives with respect to x, y and z.
protected final int	t132	Derivatives with respect to x, y and z.
protected final int	t140	Derivatives with respect to x and y.
protected final int	t141	Derivatives with respect to x, y and z.
protected final int	t150	Derivatives with respect to x and y.
protected final int	t200	Second derivative with respect to x.
protected final int	t201	Derivatives with respect to x and z.
protected final int	t202	Derivatives with respect to x and z.
protected final int	t203	Derivatives with respect to x and z.
protected final int	t204	Derivatives with respect to x and z.
protected final int	t210	Derivatives with respect to x and y.
protected final int	t211	Derivatives with respect to x, y and z.
protected final int	t212	Derivatives with respect to x, y and z.
protected final int	t213	Derivatives with respect to x, y and z.
protected final int	t220	Derivatives with respect to x and y.
protected final int	t221	Derivatives with respect to x, y and z.
protected final int	t222	Derivatives with respect to x, y and z.
protected final int	t230	Derivatives with respect to x and y.
protected final int	t231	Derivatives with respect to x, y and z.
protected final int	t240	Derivatives with respect to x and y.
protected final int	t300	Third derivative with respect to x.
protected final int	t301	Derivatives with respect to x and z.
protected final int	t302	Derivatives with respect to x and z.
protected final int	t303	Derivatives with respect to x and z.
protected final int	t310	Derivatives with respect to x and y.
protected final int	t311	Derivatives with respect to x, y and z.
protected final int	t312	Derivatives with respect to x, y and z.
protected final int	t320	Derivatives with respect to x and y.
protected final int	t321	Derivatives with respect to x, y and z.
protected final int	t330	Derivatives with respect to x and y.
protected final int	t400	Fourth derivative with respect to x.
protected final int	t401	Derivatives with respect to x and z.
protected final int	t402	Derivatives with respect to x and z.
protected final int	t410	Derivatives with respect to x and y.
protected final int	t411	Derivatives with respect to x, y and z.
protected final int	t420	Derivatives with respect to x and y.
protected final int	t500	Fifth derivative with respect to x.
protected final int	t501	Derivatives with respect to x and z.
protected final int	t510	Derivatives with respect to x and y.
protected final int	t600	Sixth derivative with respect to x.
protected final double[]	work	Store the work array to avoid memory consumption.
protected double	x	Xk - Xi.
protected double	y	Yk - Yi.
protected double	z	Zk - Zi.

Constructor Summary

Constructors

Constructor	Description
MultipoleTensor(CoordinateSystem coordinates, int order)	Constructor for MultipoleTensor.

Method Summary

Modifier and Type	Method	Description
protected abstract void	chargeIPotentialAtK(PolarizableMultipole mI, int order)	Compute the field components due to charge I at site K.
protected abstract void	chargeKPotentialAtI(PolarizableMultipole mK, int order)	Compute the field components due to multipole K at site I.
protected void	codePotentialMultipoleI(PolarizableMultipole mI, double[] T, int l, int m, int n, StringBuilder sb)	Collect the potential its partial derivatives at K due to multipole moments at the origin.
protected void	codePotentialMultipoleISIMD(PolarizableMultipole mI, double[] T, int l, int m, int n, StringBuilder sb)	Collect the potential its partial derivatives at K due to multipole moments at the origin using SIMD instructions.
protected void	codePotentialMultipoleK(PolarizableMultipole mK, double[] T, int l, int m, int n, StringBuilder sb)	Collect the potential its partial derivatives at the origin due to multipole moments at site K.
protected void	codePotentialMultipoleKSIMD(PolarizableMultipole mK, double[] T, int l, int m, int n, StringBuilder sb)	Collect the potential its partial derivatives at the origin due to multipole moments at site K using SIMD instructions.
protected abstract String	codeTensorRecursion(double[] r, double[] tensor)	This function is a driver to collect elements of the Cartesian multipole tensor.
protected double	contractMultipoleI(PolarizableMultipole mI, double[] T, int l, int m, int n)	Contract multipole moments with their respective electrostatic potential derivatives.
protected abstract void	dipoleIPotentialAtK(double uxi, double uyi, double uzi, int order)	Compute the induced dipole field components due to site I at site K.
protected abstract void	dipoleKPotentialAtI(double uxk, double uyk, double uzk, int order)	Compute the induced dipole field components due to site K at site I.
protected final void	dipoleTorque(PolarizableMultipole m, double[] torque)	Compute the torque on a permanent dipole.
void	generateTensor()	Generate the tensor using hard-coded methods or via recursion.
final void	generateTensor(double[] r)	For the MultipoleTensorTest class and testing.
double	getd2EdZ2()	getd2EdZ2.
double	getdEdZ()	getdEdZ.
protected final void	getTensor(double[] T)	Load the tensor components.
void	log(double[] tensor)	log.
protected final double	multipoleEnergy(PolarizableMultipole m)	Contract a multipole with the potential and its derivatives.
double	multipoleEnergy(PolarizableMultipole mI, PolarizableMultipole mK)	Permanent multipole energy.
double	multipoleEnergyAndGradient(PolarizableMultipole mI, PolarizableMultipole mK, double[] Gi, double[] Gk, double[] Ti, double[] Tk)	Permanent multipole energy and gradient.
protected final void	multipoleGradient(PolarizableMultipole m, double[] g)	Compute the permanent multipole gradient.
protected abstract void	multipoleIPotentialAtK(PolarizableMultipole mI, int order)	Compute the field components due to multipole I at site K.
protected abstract void	multipoleKPotentialAtI(PolarizableMultipole mK, int order)	Compute the field components due to multipole K at site I.
protected final void	multipoleTorque(PolarizableMultipole m, double[] torque)	Compute the torque on a permanent multipole.
protected abstract void	noStorageRecursion(double[] tensor)	This method is a driver to collect elements of the Cartesian multipole tensor given the recursion relationships implemented by the method "Tlmnj", which can be called directly to get a single tensor element.
protected abstract void	noStorageRecursion(double[] r, double[] tensor)	This method is a driver to collect elements of the Cartesian multipole tensor given the recursion relationships implemented by the method "Tlmnj", which can be called directly to get a single tensor element.
protected abstract void	order1()	Hard coded computation of the Cartesian multipole tensors up to 1st order.
protected abstract void	order2()	Hard coded computation of the Cartesian multipole tensors up to 2nd order.
protected abstract void	order3()	Hard coded computation of the Cartesian multipole tensors up to 3rd order.
protected abstract void	order4()	Hard coded computation of the Cartesian multipole tensors up to 4th order.
protected abstract void	order5()	Hard coded computation of the Cartesian multipole tensors up to 5th order, which is needed for quadrupole-quadrupole forces.
protected abstract void	order6()	Hard coded computation of the Cartesian multipole tensors up to 6th order, which is needed for quadrupole-quadrupole forces and orthogonal space sampling.
protected final double	polarizationEnergy(PolarizableMultipole m)	Contract an induced dipole with the potential and its derivatives.
double	polarizationEnergy(PolarizableMultipole mI, PolarizableMultipole mK, double scaleEnergy)	Polarization Energy.
double	polarizationEnergyAndGradient(PolarizableMultipole mI, PolarizableMultipole mK, double inductionMask, double energyMask, double mutualMask, double[] Gi, double[] Ti, double[] Tk)	Polarization Energy and Gradient.
protected final double	polarizationEnergyS(PolarizableMultipole m)	Contract an induced dipole with the potential and its derivatives.
protected final void	potentialMultipoleI(PolarizableMultipole mI, double[] T, int l, int m, int n)	Collect the field at R due to Q multipole moments at the origin (site I).
protected abstract void	quadrupoleIPotentialAtK(PolarizableMultipole mI, int order)	Compute the field components due to quadrupole I at site K.
protected abstract void	quadrupoleKPotentialAtI(PolarizableMultipole mK, int order)	Compute the field components due to multipole K at site I.
protected final void	quadrupoleTorque(PolarizableMultipole m, double[] torque)	Compute the torque on a permanent quadrupole.
protected abstract void	recursion(double[] tensor)	This method is a driver to collect elements of the Cartesian multipole tensor using recursion relationships and storing intermediate values.
protected abstract void	recursion(double[] r, double[] tensor)	This method is a driver to collect elements of the Cartesian multipole tensor using recursion relationships and storing intermediate values.
final void	setR(double[] r)	Set the separation vector.
abstract void	setR(double dx, double dy, double dz)	Set the separation vector.
protected final void	setTensor(double[] T)	Set the tensor components.
protected abstract void	source(double[] T000)	Generate source terms for the Challacombe et al. recursion.
protected final int	ti(int dx, int dy, int dz)	The index is based on the idea of filling tetrahedron.
protected abstract double	Tlmnj(int l, int m, int n, int j, double[] r, double[] T000)	This routine implements the recurrence relations for computation of any Cartesian multipole tensor in ~O(L^8) time, where L is the total order l + m + n, given the auxiliary elements T0000.
double	totalEnergy(PolarizableMultipole mI, PolarizableMultipole mK, double scaleEnergy, double[] energyComponents)	Permanent Multipole + Polarization Energy.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Details

order

protected final int order

Order of the tensor recursion (5th is needed for AMOEBA forces).

o1

protected final int o1

Order plus 1.

il

protected final int il

Order plus one.

im

protected final int im

im = (Order plus one)^2.

in

protected final int in

in = (Order plus one)^3.

size

protected final int size

Size = (order + 1) * (order + 2) * (order + 3) / 6;

operator

protected Operator operator

The OPERATOR in use.

coulombSource

protected final double[] coulombSource

These are the "source" terms for the recursion for the Coulomb operator (1/R).

work

protected final double[] work

Store the work array to avoid memory consumption. Note that rather than use an array for intermediate values, a 4D matrix was tried. It was approximately 50% slower than the linear work array.

T000

protected final double[] T000

Store the auxiliary tensor memory to avoid memory consumption.

coordinates

protected final CoordinateSystem coordinates

The coordinate system in use (global or QI).

R

protected double R

Separation distance.

r2

protected double r2

Separation distance squared.

x

protected double x

Xk - Xi.

y

protected double y

Yk - Yi.

z

protected double z

Zk - Zi.

t000

protected final int t000

No derivatives.

t100

protected final int t100

First derivative with respect to x.

t010

protected final int t010

First derivative with respect to y.

t001

protected final int t001

First derivative with respect to z.

t200

protected final int t200

Second derivative with respect to x.

t020

protected final int t020

Second derivative with respect to y.

t002

protected final int t002

Second derivative with respect to z.

t110

protected final int t110

Derivatives with respect to x and y.

t101

protected final int t101

Derivatives with respect to x and z.

t011

protected final int t011

Derivatives with respect to y and z.

t300

protected final int t300

Third derivative with respect to x.

t030

protected final int t030

Third derivative with respect to y.

t003

protected final int t003

Third derivative with respect to z.

t210

protected final int t210

Derivatives with respect to x and y.

t201

protected final int t201

Derivatives with respect to x and z.

t120

protected final int t120

Derivatives with respect to x and y.

t021

protected final int t021

Derivatives with respect to y and z.

t102

protected final int t102

Derivatives with respect to x and z.

t012

protected final int t012

Derivatives with respect to y and z.

t111

protected final int t111

Derivatives with respect to x, y and z.

t400

protected final int t400

Fourth derivative with respect to x.

t040

protected final int t040

Fourth derivative with respect to y.

t004

protected final int t004

Fourth derivative with respect to z.

t310

protected final int t310

Derivatives with respect to x and y.

t301

protected final int t301

Derivatives with respect to x and z.

t130

protected final int t130

Derivatives with respect to x and y.

t031

protected final int t031

Derivatives with respect to y and z.

t103

protected final int t103

Derivatives with respect to x and z.

t013

protected final int t013

Derivatives with respect to y and z.

t220

protected final int t220

Derivatives with respect to x and y.

t202

protected final int t202

Derivatives with respect to x and z.

t022

protected final int t022

Derivatives with respect to y and z.

t211

protected final int t211

Derivatives with respect to x, y and z.

t121

protected final int t121

Derivatives with respect to x, y and z.

t112

protected final int t112

Derivatives with respect to x, y and z.

t500

protected final int t500

Fifth derivative with respect to x.

t050

protected final int t050

Fifth derivative with respect to y.

t005

protected final int t005

Fifth derivative with respect to z.

t410

protected final int t410

Derivatives with respect to x and y.

t401

protected final int t401

Derivatives with respect to x and z.

t140

protected final int t140

Derivatives with respect to x and y.

t041

protected final int t041

Derivatives with respect to y and z.

t104

protected final int t104

Derivatives with respect to x and z.

t014

protected final int t014

Derivatives with respect to y and z.

t320

protected final int t320

Derivatives with respect to x and y.

t302

protected final int t302

Derivatives with respect to x and z.

t230

protected final int t230

Derivatives with respect to x and y.

t032

protected final int t032

Derivatives with respect to y and z.

t203

protected final int t203

Derivatives with respect to x and z.

t023

protected final int t023

Derivatives with respect to y and z.

t311

protected final int t311

Derivatives with respect to x, y and z.

t131

protected final int t131

Derivatives with respect to x, y and z.

t113

protected final int t113

Derivatives with respect to x, y and z.

t221

protected final int t221

Derivatives with respect to x, y and z.

t212

protected final int t212

Derivatives with respect to x, y and z.

t122

protected final int t122

Derivatives with respect to x, y and z.

t600

protected final int t600

Sixth derivative with respect to x.

t060

protected final int t060

Sixth derivative with respect to y.

t006

protected final int t006

Sixth derivative with respect to z.

t510

protected final int t510

Derivatives with respect to x and y.

t501

protected final int t501

Derivatives with respect to x and z.

t150

protected final int t150

Derivatives with respect to x and y.

t051

protected final int t051

Derivatives with respect to y and z.

t105

protected final int t105

Derivatives with respect to x and z.

t015

protected final int t015

Derivatives with respect to y and z.

t420

protected final int t420

Derivatives with respect to x and y.

t402

protected final int t402

Derivatives with respect to x and z.

t240

protected final int t240

Derivatives with respect to x and y.

t042

protected final int t042

Derivatives with respect to y and z.

t204

protected final int t204

Derivatives with respect to x and z.

t024

protected final int t024

Derivatives with respect to y and z.

t411

protected final int t411

Derivatives with respect to x, y and z.

t141

protected final int t141

Derivatives with respect to x, y and z.

t114

protected final int t114

Derivatives with respect to x, y and z.

t330

protected final int t330

Derivatives with respect to x and y.

t303

protected final int t303

Derivatives with respect to x and z.

t033

protected final int t033

Derivatives with respect to y and z.

t321

protected final int t321

Derivatives with respect to x, y and z.

t231

protected final int t231

Derivatives with respect to x, y and z.

t213

protected final int t213

Derivatives with respect to x, y and z.

t312

protected final int t312

Derivatives with respect to x, y and z.

t132

protected final int t132

Derivatives with respect to x, y and z.

t123

protected final int t123

Derivatives with respect to x, y and z.

t222

protected final int t222

Derivatives with respect to x, y and z.

Constructor Details

MultipoleTensor

public MultipoleTensor(CoordinateSystem coordinates,
 int order)

Constructor for MultipoleTensor.

Parameters:

coordinates - a CoordinateSystem object.

order - The order of the tensor.

Method Details

getd2EdZ2

public double getd2EdZ2()

getd2EdZ2.

Returns:

a double.

getdEdZ

public double getdEdZ()

getdEdZ.

Returns:

a double.

log

public void log(double[] tensor)

log.

Parameters:

tensor - an array of double values.

multipoleEnergy

public double multipoleEnergy(PolarizableMultipole mI,
 PolarizableMultipole mK)

Permanent multipole energy.

Parameters:

mI - PolarizableMultipole at site I.

mK - PolarizableMultipole at site K.

Returns:

a double.

multipoleEnergyAndGradient

public double multipoleEnergyAndGradient(PolarizableMultipole mI,
 PolarizableMultipole mK,
 double[] Gi,
 double[] Gk,
 double[] Ti,
 double[] Tk)

Permanent multipole energy and gradient.

Parameters:

mI - PolarizableMultipole at site I.

mK - PolarizableMultipole at site K.

Gi - Coordinate gradient at site I.

Gk - Coordinate gradient at site K.

Ti - Torque at site I.

Tk - Torque at site K.

Returns:

the permanent multipole energy.

polarizationEnergy

public double polarizationEnergy(PolarizableMultipole mI,
 PolarizableMultipole mK,
 double scaleEnergy)

Polarization Energy.

Parameters:

mI - PolarizableMultipole at site I.

mK - PolarizableMultipole at site K.

scaleEnergy - a double.

Returns:

a double.

polarizationEnergyAndGradient

public double polarizationEnergyAndGradient(PolarizableMultipole mI,
 PolarizableMultipole mK,
 double inductionMask,
 double energyMask,
 double mutualMask,
 double[] Gi,
 double[] Ti,
 double[] Tk)

Polarization Energy and Gradient.

Parameters:

mI - PolarizableMultipole at site I.

mK - PolarizableMultipole at site K.

inductionMask - a double.

energyMask - a double.

mutualMask - a double.

Gi - an array of double values.

Ti - an array of double values.

Tk - an array of double values.

Returns:

a double.

totalEnergy

public double totalEnergy(PolarizableMultipole mI,
 PolarizableMultipole mK,
 double scaleEnergy,
 double[] energyComponents)

Permanent Multipole + Polarization Energy.

Parameters:

mI - PolarizableMultipole at site I.

mK - PolarizableMultipole at site K.

scaleEnergy - a double.

energyComponents - The permanent and induced energy components.

Returns:

a double.

setR

public final void setR(double[] r)

Set the separation vector.

Parameters:

r - The separation vector.

setR

public abstract void setR(double dx,
 double dy,
 double dz)

Set the separation vector.

Parameters:

dx - Separation along the X-axis.

dy - Separation along the Y-axis.

dz - Separation along the Z-axis.

generateTensor

public final void generateTensor(double[] r)

For the MultipoleTensorTest class and testing.

Parameters:

r - an array of double values.

generateTensor

public void generateTensor()

Generate the tensor using hard-coded methods or via recursion.

source

protected abstract void source(double[] T000)

Generate source terms for the Challacombe et al. recursion.

Parameters:

T000 - Location to store the source terms.

ti

protected final int ti(int dx,
 int dy,
 int dz)

The index is based on the idea of filling tetrahedron.

1/r has an index of 0.
derivatives of x are first; indices from 1..o for d/dx..(d/dx)^o
derivatives of x and y are second; base triangle of size (o+1)(o+2)/2
derivatives of x, y and z are last; total size (o+1)*(o+2)*(o+3)/6

This function is useful to set up masking constants:
static int Tlmn = ti(l,m,n,order)
For example the (d/dy)^2 (1/R) storage location:
static int T020 = ti(0,2,0,order)

Parameters:

dx - int The number of d/dx operations.

dy - int The number of d/dy operations.

dz - int The number of d/dz operations.

Returns:

int in the range (0..binomial(order + 3, 3) - 1)

contractMultipoleI

protected double contractMultipoleI(PolarizableMultipole mI,
 double[] T,
 int l,
 int m,
 int n)

Contract multipole moments with their respective electrostatic potential derivatives.

Parameters:

mI - PolarizableMultipole at site I.

T - array of electrostatic potential and partial derivatives

l - apply (d/dx)^l to the potential

m - apply (d/dy)^l to the potential

n - apply (d/dz)^l to the potential

Returns:

the contracted interaction.

potentialMultipoleI

protected final void potentialMultipoleI(PolarizableMultipole mI,
 double[] T,
 int l,
 int m,
 int n)

Collect the field at R due to Q multipole moments at the origin (site I).

Parameters:

mI - PolarizableMultipole at site I.

T - Electrostatic potential and partial derivatives

l - apply (d/dx)^l to the potential

m - apply (d/dy)^l to the potential

n - apply (d/dz)^l to the potential

codeTensorRecursion

protected abstract String codeTensorRecursion(double[] r,
 double[] tensor)

This function is a driver to collect elements of the Cartesian multipole tensor. Collecting all tensors scales slightly better than O(order^4).

For a multipole expansion truncated at quadrupole order, for example, up to order 5 is needed for energy gradients. The number of terms this requires is binomial(5 + 3, 3) or 8! / (5! * 3!), which is 56.

The packing of the tensor elements for order = 1
tensor[0] = 1/|r|
tensor[1] = -x/|r|^3
tensor[2] = -y/|r|^3
tensor[3] = -z/|r|^3

Parameters:

r - The separation vector.

tensor - The tensor elements.

Returns:

The tensor recursion code.

Since:

1.0

codePotentialMultipoleI

protected void codePotentialMultipoleI(PolarizableMultipole mI,
 double[] T,
 int l,
 int m,
 int n,
 StringBuilder sb)

Collect the potential its partial derivatives at K due to multipole moments at the origin.

Parameters:

mI - PolarizableMultipole at site I.

T - Electrostatic potential and partial derivatives.

l - apply (d/dx)^l to the potential.

m - apply (d/dy)^l to the potential.

n - apply (d/dz)^l to the potential.

sb - Append the code to the StringBuilder.

codePotentialMultipoleISIMD

protected void codePotentialMultipoleISIMD(PolarizableMultipole mI,
 double[] T,
 int l,
 int m,
 int n,
 StringBuilder sb)

Collect the potential its partial derivatives at K due to multipole moments at the origin using SIMD instructions.

Parameters:

mI - PolarizableMultipole at site I.

T - Electrostatic potential and partial derivatives.

l - apply (d/dx)^l to the potential.

m - apply (d/dy)^l to the potential.

n - apply (d/dz)^l to the potential.

sb - Append the code to the StringBuilder.

codePotentialMultipoleK

protected void codePotentialMultipoleK(PolarizableMultipole mK,
 double[] T,
 int l,
 int m,
 int n,
 StringBuilder sb)

Collect the potential its partial derivatives at the origin due to multipole moments at site K.

Parameters:

mK - PolarizableMultipole at site I.

T - Electrostatic potential and partial derivatives.

l - apply (d/dx)^l to the potential.

m - apply (d/dy)^l to the potential.

n - apply (d/dz)^l to the potential.

sb - Append the code to the StringBuilder.

codePotentialMultipoleKSIMD

protected void codePotentialMultipoleKSIMD(PolarizableMultipole mK,
 double[] T,
 int l,
 int m,
 int n,
 StringBuilder sb)

Collect the potential its partial derivatives at the origin due to multipole moments at site K using SIMD instructions.

Parameters:

mK - PolarizableMultipole at site I.

T - Electrostatic potential and partial derivatives.

l - apply (d/dx)^l to the potential.

m - apply (d/dy)^l to the potential.

n - apply (d/dz)^l to the potential.

sb - Append the code to the StringBuilder.

multipoleEnergy

protected final double multipoleEnergy(PolarizableMultipole m)

Contract a multipole with the potential and its derivatives.

Parameters:

m - PolarizableMultipole at the site of the potential.

Returns:

The permanent multipole energy.

multipoleGradient

protected final void multipoleGradient(PolarizableMultipole m,
 double[] g)

Compute the permanent multipole gradient.

Parameters:

m - PolarizableMultipole at the site of the potential.

g - The atomic gradient.

multipoleTorque

protected final void multipoleTorque(PolarizableMultipole m,
 double[] torque)

Compute the torque on a permanent multipole.

Parameters:

m - PolarizableMultipole at the site of the potential.

torque - an array of double values.

dipoleTorque

protected final void dipoleTorque(PolarizableMultipole m,
 double[] torque)

Compute the torque on a permanent dipole.

Parameters:

m - PolarizableMultipole at the site of the potential.

torque - an array of double values.

quadrupoleTorque

protected final void quadrupoleTorque(PolarizableMultipole m,
 double[] torque)

Compute the torque on a permanent quadrupole.

Parameters:

m - PolarizableMultipole at the site of the potential.

torque - an array of double values.

polarizationEnergy

protected final double polarizationEnergy(PolarizableMultipole m)

Contract an induced dipole with the potential and its derivatives.

Parameters:

m - PolarizableMultipole at the site of the potential.

Returns:

The polarization energy.

polarizationEnergyS

protected final double polarizationEnergyS(PolarizableMultipole m)

Contract an induced dipole with the potential and its derivatives.

Parameters:

m - PolarizableMultipole at the site of the potential.

Returns:

The polarization energy.

getTensor

protected final void getTensor(double[] T)

Load the tensor components.

Parameters:

T - an array of double values.

setTensor

protected final void setTensor(double[] T)

Set the tensor components.

Parameters:

T - an array of double values.

noStorageRecursion

protected abstract void noStorageRecursion(double[] tensor)

This method is a driver to collect elements of the Cartesian multipole tensor given the recursion relationships implemented by the method "Tlmnj", which can be called directly to get a single tensor element. It does not store intermediate values of the recursion, causing it to scale O(order^8). For order = 5, this approach is a factor of 10 slower than recursion.

Parameters:

tensor - double[] length must be at least binomial(order + 3, 3).

noStorageRecursion

protected abstract void noStorageRecursion(double[] r,
 double[] tensor)

This method is a driver to collect elements of the Cartesian multipole tensor given the recursion relationships implemented by the method "Tlmnj", which can be called directly to get a single tensor element. It does not store intermediate values of the recursion, causing it to scale O(order^8). For order = 5, this approach is a factor of 10 slower than recursion.

Parameters:

r - double[] vector between two sites.

tensor - double[] length must be at least binomial(order + 3, 3).

Tlmnj

protected abstract double Tlmnj(int l,
 int m,
 int n,
 int j,
 double[] r,
 double[] T000)

This routine implements the recurrence relations for computation of any Cartesian multipole tensor in ~O(L^8) time, where L is the total order l + m + n, given the auxiliary elements T0000.
It implements the recursion relationships in brute force fashion, without saving intermediate values. This is useful for finding a single tensor, rather than all binomial(L + 3, 3).
The specific recursion equations (41-43) and set of auxiliary tensor elements from equation (40) can be found in Challacombe et al.

Parameters:

l - int The number of (d/dx) operations.

m - int The number of (d/dy) operations.

n - int The number of (d/dz) operations.

j - int j = 0 is the Tlmn tensor, j .GT. 0 is an intermediate.

r - double[] The {x,y,z} coordinates.

T000 - double[] Initial auxiliary tensor elements from Eq. (40).

Returns:

double The requested Tensor element (intermediate if j .GT. 0).

Since:

1.0

recursion

protected abstract void recursion(double[] tensor)

This method is a driver to collect elements of the Cartesian multipole tensor using recursion relationships and storing intermediate values. It scales approximately O(order^4).

Parameters:

tensor - double[] length must be at least binomial(order + 3, 3).

recursion

protected abstract void recursion(double[] r,
 double[] tensor)

This method is a driver to collect elements of the Cartesian multipole tensor using recursion relationships and storing intermediate values. It scales approximately O(order^4).

Parameters:

r - double[] vector between two sites.

tensor - double[] length must be at least binomial(order + 3, 3).

order1

protected abstract void order1()

Hard coded computation of the Cartesian multipole tensors up to 1st order.

order2

protected abstract void order2()

Hard coded computation of the Cartesian multipole tensors up to 2nd order.

order3

protected abstract void order3()

Hard coded computation of the Cartesian multipole tensors up to 3rd order.

order4

protected abstract void order4()

Hard coded computation of the Cartesian multipole tensors up to 4th order.

order5

protected abstract void order5()

Hard coded computation of the Cartesian multipole tensors up to 5th order, which is needed for quadrupole-quadrupole forces.

order6

protected abstract void order6()

Hard coded computation of the Cartesian multipole tensors up to 6th order, which is needed for quadrupole-quadrupole forces and orthogonal space sampling.

multipoleIPotentialAtK

protected abstract void multipoleIPotentialAtK(PolarizableMultipole mI,
 int order)

Compute the field components due to multipole I at site K.

Parameters:

mI - PolarizableMultipole at site I.

order - Compute derivatives of the potential up to this order. Order 0: Electrostatic potential (E000) Order 1: First derivatives: d/dX is E100, d/dY is E010, d/dZ is E001. Order 2: Second derivatives: d2/dXdX is E200, d2/dXdY is E110 (needed for quadrupole energy) Order 3: 3rd derivatives: (needed for quadrupole forces).

chargeIPotentialAtK

protected abstract void chargeIPotentialAtK(PolarizableMultipole mI,
 int order)

Compute the field components due to charge I at site K.

Parameters:

mI - PolarizableMultipole at site I.

order - Compute derivatives of the potential up to this order. Order 0: Electrostatic potential (E000) Order 1: First derivatives: d/dX is E100, d/dY is E010, d/dZ is E001. Order 2: Second derivatives: d2/dXdX is E200, d2/dXdY is E110 (needed for quadrupole energy) Order 3: 3rd derivatives: (needed for quadrupole forces).

dipoleIPotentialAtK

protected abstract void dipoleIPotentialAtK(double uxi,
 double uyi,
 double uzi,
 int order)

Compute the induced dipole field components due to site I at site K.

Parameters:

uxi - X-dipole component.

uyi - Y-dipole component.

uzi - Z-dipole component.

order - Potential order.

quadrupoleIPotentialAtK

protected abstract void quadrupoleIPotentialAtK(PolarizableMultipole mI,
 int order)

Compute the field components due to quadrupole I at site K.

Parameters:

mI - PolarizableMultipole at site I.

order - Compute derivatives of the potential up to this order. Order 0: Electrostatic potential (E000) Order 1: First derivatives: d/dX is E100, d/dY is E010, d/dZ is E001. Order 2: Second derivatives: d2/dXdX is E200, d2/dXdY is E110 (needed for quadrupole energy) Order 3: 3rd derivatives: (needed for quadrupole forces).

multipoleKPotentialAtI

protected abstract void multipoleKPotentialAtI(PolarizableMultipole mK,
 int order)

Compute the field components due to multipole K at site I.

Parameters:

mK - PolarizableMultipole at site K.

order - Compute derivatives of the potential up to this order. Order 0: Electrostatic potential (E000) Order 1: First derivatives: d/dX is E100, d/dY is E010, d/dZ is E001. Order 2: Second derivatives: d2/dXdX is E200, d2/dXdY is E110 (needed for quadrupole energy) Order 3: 3rd derivatives: (needed for quadrupole forces).

chargeKPotentialAtI

protected abstract void chargeKPotentialAtI(PolarizableMultipole mK,
 int order)

Compute the field components due to multipole K at site I.

Parameters:

mK - PolarizableMultipole at site K.

order - Compute derivatives of the potential up to this order. Order 0: Electrostatic potential (E000) Order 1: First derivatives: d/dX is E100, d/dY is E010, d/dZ is E001. Order 2: Second derivatives: d2/dXdX is E200, d2/dXdY is E110 (needed for quadrupole energy) Order 3: 3rd derivatives: (needed for quadrupole forces).

dipoleKPotentialAtI

protected abstract void dipoleKPotentialAtI(double uxk,
 double uyk,
 double uzk,
 int order)

Compute the induced dipole field components due to site K at site I.

Parameters:

uxk - X-dipole component.

uyk - Y-dipole component.

uzk - Z-dipole component.

order - Potential order.

quadrupoleKPotentialAtI

protected abstract void quadrupoleKPotentialAtI(PolarizableMultipole mK,
 int order)

Compute the field components due to multipole K at site I.

Parameters:

mK - PolarizableMultipole at site K.

order - Compute derivatives of the potential up to this order. Order 0: Electrostatic potential (E000) Order 1: First derivatives: d/dX is E100, d/dY is E010, d/dZ is E001. Order 2: Second derivatives: d2/dXdX is E200, d2/dXdY is E110 (needed for quadrupole energy) Order 3: 3rd derivatives: (needed for quadrupole forces).