# ewald (c39b2)

The Ewald Summation method

Invoking the Ewald summation for calculating the electrostatic interactions

can be specified any time the nbond specification parser is invoked. See

the syntax section for a list of all commands that invoke this parser.

Prerequisite reading: » nbonds

* Syntax | Syntax of the Ewald summation specification

* Defaults | Defaults used in the specification

* Function | Description of the options

* Discussion | More general discussion of the algorithm

Invoking the Ewald summation for calculating the electrostatic interactions

can be specified any time the nbond specification parser is invoked. See

the syntax section for a list of all commands that invoke this parser.

Prerequisite reading: » nbonds

* Syntax | Syntax of the Ewald summation specification

* Defaults | Defaults used in the specification

* Function | Description of the options

* Discussion | More general discussion of the algorithm

Top

[SYNTAX EWALD]

{ NBONds } { nonbond-spec }

{ UPDAte } { }

{ ENERgy } { }

{ MINImize } { }

{ DYNAmics } { }

The keywords are:

nonbond-spec::= [ method-spec ]

{ [ NOEWald ] }

{ }

method-spec::= { EWALd [ewald-spec] { [ NOPMewald [std-ew-spec] ] } }

{ { PMEWald [pmesh-spec] } }

ewald-spec::= KAPPa real [erfc-spec]

std-ew-spec::= { [ KMAX integer ] } KSQMAX integer

{ KMXX integer KMXY integer KMXZ integer }

pmesh-spec::= FFTX int FFTY int FFTZ int ORDEr integer [QCOR real (***) ]

erfc-spec::= { SPLIne { [EWMIn real] [EWMAx real] [EWNPts int] } }

{ INTErpolate { } }

{ }

{ ABROmowitz }

{ CHEBychev }

{ EXACt_high_precision }

{ LOWPrecision_exact }

{ ERFMode int }

[SYNTAX EWALD]

{ NBONds } { nonbond-spec }

{ UPDAte } { }

{ ENERgy } { }

{ MINImize } { }

{ DYNAmics } { }

The keywords are:

nonbond-spec::= [ method-spec ]

{ [ NOEWald ] }

{ }

method-spec::= { EWALd [ewald-spec] { [ NOPMewald [std-ew-spec] ] } }

{ { PMEWald [pmesh-spec] } }

ewald-spec::= KAPPa real [erfc-spec]

std-ew-spec::= { [ KMAX integer ] } KSQMAX integer

{ KMXX integer KMXY integer KMXZ integer }

pmesh-spec::= FFTX int FFTY int FFTZ int ORDEr integer [QCOR real (***) ]

erfc-spec::= { SPLIne { [EWMIn real] [EWMAx real] [EWNPts int] } }

{ INTErpolate { } }

{ }

{ ABROmowitz }

{ CHEBychev }

{ EXACt_high_precision }

{ LOWPrecision_exact }

{ ERFMode int }

Top

The defaults for the ewald summation are set internally

and are currently set to NOEWald, KAPPa=1.0, KMAX=5, KSQMax=27, and

NOPMewald, KAPPa=1.0, FFTX=FFTY=FFTZ=32, ORDEr=4, QCOR=1.0

Recommended values for Ewald are:

EWALD PMEWald KAPPa 0.34 ORDEr 6 -

FFTX intboxvx FFTY intboxvy FFTZ intboxvz -

CTOFNB 12.0 CUTNB 14.0 QCOR 1.0(***)

Where intboxv* is an integer value similar to or larger than the corresponding

unit cell dimension that has prime factors of 2,3, and 5 only (2,3 preferred).

grid point spacing should be between 0.8 and 1.2 Angstroms.

These recommended values should give relative force errors of roughly 10**-5.

To reduce the total PME cost at the expense of accuracy, decrease the cutoff

distances while increasing KAPPa (keep the product near 4) reduces the real

space cost. To reduce the K-space cost, either reduce ORDEr from 6 to 4 or

increase the grid spacing up to perhaps 1.5 Angstroms.

(***) The QCOR value should be 1.0 for vacuum, solid, or finite systems.

For periodic systems in solution, it should be reduced (or set to zero) by an

amount that depends on how the net charge is distributed and on the effective

dielectric constant. For a treatise on this correction term, see:

S. Bogusz, T. Cheatham, and B. Brooks, JCP (1998) 108, 7070-7084 and references

contained therein (esp. Hummer and Levy).

The defaults for the ewald summation are set internally

and are currently set to NOEWald, KAPPa=1.0, KMAX=5, KSQMax=27, and

NOPMewald, KAPPa=1.0, FFTX=FFTY=FFTZ=32, ORDEr=4, QCOR=1.0

Recommended values for Ewald are:

EWALD PMEWald KAPPa 0.34 ORDEr 6 -

FFTX intboxvx FFTY intboxvy FFTZ intboxvz -

CTOFNB 12.0 CUTNB 14.0 QCOR 1.0(***)

Where intboxv* is an integer value similar to or larger than the corresponding

unit cell dimension that has prime factors of 2,3, and 5 only (2,3 preferred).

grid point spacing should be between 0.8 and 1.2 Angstroms.

These recommended values should give relative force errors of roughly 10**-5.

To reduce the total PME cost at the expense of accuracy, decrease the cutoff

distances while increasing KAPPa (keep the product near 4) reduces the real

space cost. To reduce the K-space cost, either reduce ORDEr from 6 to 4 or

increase the grid spacing up to perhaps 1.5 Angstroms.

(***) The QCOR value should be 1.0 for vacuum, solid, or finite systems.

For periodic systems in solution, it should be reduced (or set to zero) by an

amount that depends on how the net charge is distributed and on the effective

dielectric constant. For a treatise on this correction term, see:

S. Bogusz, T. Cheatham, and B. Brooks, JCP (1998) 108, 7070-7084 and references

contained therein (esp. Hummer and Levy).

Top

i) The EWALD keyword invokes the Ewald summation for calculation of

electrostatic interactions in periodic, neutral systems. The formulation of

the Ewald summation dictates that the primary system must be neutral. If

otherwise, the summation is not formally correct and some

convergence problems may result. The NOEWald (default) suppresses the Ewald

method for calculating electrostatic interactions. Van der waals

options VSHIFT and VSWITCH are supported with ewald. The algorithm

currently supports the atom and group nonbond lists and the CRYSTAL facilty

must be used. The PMEWald keyword invokes the Particle Mesh Ewald algorithm

for the reciprocal space summation. For details on the PME method, see

J. Chem. Phys. 103:8577 (1995). The EWALd algorithm is limited to CUBIC,

TETRAGONAL, and ORTHORHOMBIC unit cells. The PMEWald algorithm supports

all unit cells that are supported by the CRYSTAL facility.

ii) The KAPPa keyword, followed by a real number governs the width of the

Gaussian distribution central to the Ewald method. An approximate value

of kappa can be chosen by taking KAPPa=5/CTOFNB. This is fairly conservative.

Values of 4/CTOFNB lead to small force errors (roughly 10**-5). See

discussion section for details on choosing an optimum value of KAPPa.

iii) The KMAX key word is the number of kvectors (or images of the

primary unit cell) that will be summed in any direction. It is the

radius of the Ewald summation. For orthorombic cells, the value of

kmax may be independently specified in the x, y, and z directions with

the keywords KMXX, KMXY, and KMXZ. In the PME version, the number of

FFT grid points for the charge mesh is specified by FFTX, FFTY, and FFTZ.

iv) The KSQMax key word should be chosen between KMAX squared and 3 times

KMAX squared.

v) An appropriate, although not optimal, set of parameters can be

chosen by taking KAPPA=5/CTOFNB and KMAX=KAPPa*boxlength. The actual

values should then be performanced optimized for your particular system.

For the PME method, FFTX should be approximately the box length in Angstroms.

(for efficiency, FFTX should be a multiple of powers of 2,3, and 5).

IMPORTANT NOTE::: THE SUGGESTION THAT FFTX, FFTY, AND FFTZ HAVE

NO PRIME FACTORS OTHER THAN 2, 3, AND 5 SEEMS TO BE A REQUIREMENT.

LARGE ERRORS IN THE FORCE ARE OBSERVED WHEN THIS CONDITION IS NOT MET.

FUTURE VERSIONS OF CHARMM WILL FLAG THIS AS AN ERROR CONDITION.

ORDEr specifies the order of the B-spline interpolation, e.g. cubic is

order 4 (default), fifth degree is ORDEr 6. The ORDEr must be an even

number and at least 4.

vi) EWALd runs in parallel on both shared (PARVECT) and distributed

memory parallel computers. PME runs in parallel on distributed

memory computers.

vii) several algorithms are available for the calculation of the complimentary

error function, erfc(x). EXACt and LOWPrecision use an interative technique

described in section 6.2 of Numerical Recipies. ABRO and CHEB are polynomial

approximations. A lookup table (filled at the beginning of the simulation

using the EXACt method) can be used with either a linear (INTE) of cubic

spline (SPLINe) interpolation. SPLIne is recommended.

viii) Ewald with MMFF

A version of EWALD was developed for MMFF. The usual MMFF electrostatic

term: qq/(r+d) is split into two terms: qq/r - qq*d/(r*(r+d))

The first term is handled by the Ewald method in the usual manner

(real-space and k-space parts) and the second term is truncated

at the cutoff distance using a switching function (from CTONNB to CTOFNB).

Since the second term is quite small at the cutoff distance, the use of a

switching function should not introduce significant artificial forces.

i) The EWALD keyword invokes the Ewald summation for calculation of

electrostatic interactions in periodic, neutral systems. The formulation of

the Ewald summation dictates that the primary system must be neutral. If

otherwise, the summation is not formally correct and some

convergence problems may result. The NOEWald (default) suppresses the Ewald

method for calculating electrostatic interactions. Van der waals

options VSHIFT and VSWITCH are supported with ewald. The algorithm

currently supports the atom and group nonbond lists and the CRYSTAL facilty

must be used. The PMEWald keyword invokes the Particle Mesh Ewald algorithm

for the reciprocal space summation. For details on the PME method, see

J. Chem. Phys. 103:8577 (1995). The EWALd algorithm is limited to CUBIC,

TETRAGONAL, and ORTHORHOMBIC unit cells. The PMEWald algorithm supports

all unit cells that are supported by the CRYSTAL facility.

ii) The KAPPa keyword, followed by a real number governs the width of the

Gaussian distribution central to the Ewald method. An approximate value

of kappa can be chosen by taking KAPPa=5/CTOFNB. This is fairly conservative.

Values of 4/CTOFNB lead to small force errors (roughly 10**-5). See

discussion section for details on choosing an optimum value of KAPPa.

iii) The KMAX key word is the number of kvectors (or images of the

primary unit cell) that will be summed in any direction. It is the

radius of the Ewald summation. For orthorombic cells, the value of

kmax may be independently specified in the x, y, and z directions with

the keywords KMXX, KMXY, and KMXZ. In the PME version, the number of

FFT grid points for the charge mesh is specified by FFTX, FFTY, and FFTZ.

iv) The KSQMax key word should be chosen between KMAX squared and 3 times

KMAX squared.

v) An appropriate, although not optimal, set of parameters can be

chosen by taking KAPPA=5/CTOFNB and KMAX=KAPPa*boxlength. The actual

values should then be performanced optimized for your particular system.

For the PME method, FFTX should be approximately the box length in Angstroms.

(for efficiency, FFTX should be a multiple of powers of 2,3, and 5).

IMPORTANT NOTE::: THE SUGGESTION THAT FFTX, FFTY, AND FFTZ HAVE

NO PRIME FACTORS OTHER THAN 2, 3, AND 5 SEEMS TO BE A REQUIREMENT.

LARGE ERRORS IN THE FORCE ARE OBSERVED WHEN THIS CONDITION IS NOT MET.

FUTURE VERSIONS OF CHARMM WILL FLAG THIS AS AN ERROR CONDITION.

ORDEr specifies the order of the B-spline interpolation, e.g. cubic is

order 4 (default), fifth degree is ORDEr 6. The ORDEr must be an even

number and at least 4.

vi) EWALd runs in parallel on both shared (PARVECT) and distributed

memory parallel computers. PME runs in parallel on distributed

memory computers.

vii) several algorithms are available for the calculation of the complimentary

error function, erfc(x). EXACt and LOWPrecision use an interative technique

described in section 6.2 of Numerical Recipies. ABRO and CHEB are polynomial

approximations. A lookup table (filled at the beginning of the simulation

using the EXACt method) can be used with either a linear (INTE) of cubic

spline (SPLINe) interpolation. SPLIne is recommended.

viii) Ewald with MMFF

A version of EWALD was developed for MMFF. The usual MMFF electrostatic

term: qq/(r+d) is split into two terms: qq/r - qq*d/(r*(r+d))

The first term is handled by the Ewald method in the usual manner

(real-space and k-space parts) and the second term is truncated

at the cutoff distance using a switching function (from CTONNB to CTOFNB).

Since the second term is quite small at the cutoff distance, the use of a

switching function should not introduce significant artificial forces.

Top

The Ewald Summation in Molecular Dynamics Simulation

The electrostatic energy of a periodic system can be expressed by a lattice

sum over all pair interactions and over all lattice vectors excluding

the i=j term in the primary box. Summations carried out in this simple

way have been shown to be conditionally convergent. The method developed by

Ewald, in essence, mathematically transforms this fairly straightforward

summation to two more complicated but rapidly convergent sums. One summation

is carried out in reciporcal space while the other is carried out in real

space. Based on the formulation by Ewald, the simple lattice sum can be

reformulated to give absolutely convergent summations which define the

principal value of the electrostatic potential, called the intrinsic potential.

Given the periodicity present in both crystal calculations and in dynamics

simulations using periodic boundary conditions, the Ewald formulation becomes

well suited for the calculation of the electrostatic energy and force. If we

consider a system of point charges in the unit or primary cell, we can specify

its charge density by

ro(r) = sum_i [ q_i * delta(r-r_i)]

In the Ewald method this distribution is replaced by two other distributions

ro_1(r) = sum_i [ q_i ( delta(r-r_i) - f(r-r_i)]

and

ro_2(r) = sum_i [q_i f(r-r_i)

such that the sum of the two recovers the original. The distribution,

f(r), is a spherical distribution generally taken to be Gaussian, the

width of the gaussian dictated by the parameter, KAPPa. The charge

distributions are situated on the ion lattice positions, but integrate

to zero. The potential from the distribution ro_1(r) is a short range

potential evaluated in a direct real space summation (truncated at

CTOFNB). The diffuse charge distribution placed on the lattice sites

reduces to the potential of the corresponding point charge at large r.

ro_2(r), being a continuous distribution of Gaussians situated on the

periodic lattice positions, is a smoothly varying function of r and thus

is well approximated by a superposition of continuous functions. This

distribution is, therefore, expanded in a Fourier series and the

potential is obtained by solving the Poisson equation. The point of

splitting the problem into two parts, is that by a suitable choice of

the parameter KAPPa we can get very good convergence of both parts of

the summation.

For the real space part of the energy, we choose kappa so that the

complementary error function term, erfc(kappa*r) decreases rapidly

enough with r to make it a good approximation to take only nearest

images in the sum and neglect the value for which r > CTOFNB. The

reciprocal space sums are rapidly convergent and a spherical cutoff in k

space is applied so that the sum over k becomes a sum over {l,m,n}, with

(l**2+m**2+n**2) < or = to KSQMAX A large value of KAPPa means that the

real space sum is more rapidly convergent but the reciprocal space sum

is less rapid. In practice one chooses KAPPa to give good convergence

at the cutoff radius, CTOFNB. KMAX is then chosen to such that the

reciprocal space calculation converges. The equation (KMAX/(box

length)=KAPPa may be used as a rough guide. Optimization with respect

to the timing trade offs, ie. how much time is spent in real space vs

k-space should be performed before a lengthy production run.

The CCP5 notes in several articles in 1993 cover some possible

optimization strategies and criteria although a simple line search will

suffice. Complete optimiztion of the ewald method for a particular

application requires optimizing CTOFNB, KAPPa, and KMAX. A discussion

of optimization and error analysis can be found in Kolfka and Perram,

Molecular Simulation, 9, 351 (1992). For PME, see Feller, Pastor,

Rojnuckarin, Bogusz, and Brooks. J. Phys. Chem., 100, 42, 17011 (1996)

and some of Tom Darden's published work.

The Ewald Summation in Molecular Dynamics Simulation

The electrostatic energy of a periodic system can be expressed by a lattice

sum over all pair interactions and over all lattice vectors excluding

the i=j term in the primary box. Summations carried out in this simple

way have been shown to be conditionally convergent. The method developed by

Ewald, in essence, mathematically transforms this fairly straightforward

summation to two more complicated but rapidly convergent sums. One summation

is carried out in reciporcal space while the other is carried out in real

space. Based on the formulation by Ewald, the simple lattice sum can be

reformulated to give absolutely convergent summations which define the

principal value of the electrostatic potential, called the intrinsic potential.

Given the periodicity present in both crystal calculations and in dynamics

simulations using periodic boundary conditions, the Ewald formulation becomes

well suited for the calculation of the electrostatic energy and force. If we

consider a system of point charges in the unit or primary cell, we can specify

its charge density by

ro(r) = sum_i [ q_i * delta(r-r_i)]

In the Ewald method this distribution is replaced by two other distributions

ro_1(r) = sum_i [ q_i ( delta(r-r_i) - f(r-r_i)]

and

ro_2(r) = sum_i [q_i f(r-r_i)

such that the sum of the two recovers the original. The distribution,

f(r), is a spherical distribution generally taken to be Gaussian, the

width of the gaussian dictated by the parameter, KAPPa. The charge

distributions are situated on the ion lattice positions, but integrate

to zero. The potential from the distribution ro_1(r) is a short range

potential evaluated in a direct real space summation (truncated at

CTOFNB). The diffuse charge distribution placed on the lattice sites

reduces to the potential of the corresponding point charge at large r.

ro_2(r), being a continuous distribution of Gaussians situated on the

periodic lattice positions, is a smoothly varying function of r and thus

is well approximated by a superposition of continuous functions. This

distribution is, therefore, expanded in a Fourier series and the

potential is obtained by solving the Poisson equation. The point of

splitting the problem into two parts, is that by a suitable choice of

the parameter KAPPa we can get very good convergence of both parts of

the summation.

For the real space part of the energy, we choose kappa so that the

complementary error function term, erfc(kappa*r) decreases rapidly

enough with r to make it a good approximation to take only nearest

images in the sum and neglect the value for which r > CTOFNB. The

reciprocal space sums are rapidly convergent and a spherical cutoff in k

space is applied so that the sum over k becomes a sum over {l,m,n}, with

(l**2+m**2+n**2) < or = to KSQMAX A large value of KAPPa means that the

real space sum is more rapidly convergent but the reciprocal space sum

is less rapid. In practice one chooses KAPPa to give good convergence

at the cutoff radius, CTOFNB. KMAX is then chosen to such that the

reciprocal space calculation converges. The equation (KMAX/(box

length)=KAPPa may be used as a rough guide. Optimization with respect

to the timing trade offs, ie. how much time is spent in real space vs

k-space should be performed before a lengthy production run.

The CCP5 notes in several articles in 1993 cover some possible

optimization strategies and criteria although a simple line search will

suffice. Complete optimiztion of the ewald method for a particular

application requires optimizing CTOFNB, KAPPa, and KMAX. A discussion

of optimization and error analysis can be found in Kolfka and Perram,

Molecular Simulation, 9, 351 (1992). For PME, see Feller, Pastor,

Rojnuckarin, Bogusz, and Brooks. J. Phys. Chem., 100, 42, 17011 (1996)

and some of Tom Darden's published work.