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/* -*- c++ -*- ----------------------------------------------------------
LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
http://lammps.sandia.gov, Sandia National Laboratories
Steve Plimpton, sjplimp@sandia.gov
Copyright (2003) Sandia Corporation. Under the terms of Contract
DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
certain rights in this software. This software is distributed under
the GNU General Public License.
See the README file in the top-level LAMMPS directory.
------------------------------------------------------------------------- */
/* ----------------------------------------------------------------------
Contributing authors: Andres Jaramillo-Botero, Hai Xiao, Julius Su (Caltech)
------------------------------------------------------------------------- */
namespace LAMMPS_NS {
#define PAULI_RE 0.9
#define PAULI_RC 1.125
#define PAULI_RHO -0.2
#define ERF_TERMS1 12
#define ERF_TERMS2 7
#define DERF_TERMS 13
// error arrays
double E1[] =
{
1.483110564084803581889448079057,
-3.01071073386594942470731046311E-1,
6.8994830689831566246603180718E-2,
-1.3916271264722187682546525687E-2,
2.420799522433463662891678239E-3,
-3.65863968584808644649382577E-4,
4.8620984432319048282887568E-5,
-5.749256558035684835054215E-6,
6.11324357843476469706758E-7,
-5.8991015312958434390846E-8,
5.207009092068648240455E-9,
-4.23297587996554326810E-10,
3.1881135066491749748E-11,
-2.236155018832684273E-12,
1.46732984799108492E-13,
-9.044001985381747E-15,
5.25481371547092E-16,
-2.8874261222849E-17,
1.504785187558E-18,
-7.4572892821E-20,
3.522563810E-21,
-1.58944644E-22,
6.864365E-24,
-2.84257E-25,
1.1306E-26,
-4.33E-28,
1.6E-29,
-1.0E-30
};
double E2[] =
{
1.077977852072383151168335910348,
-2.6559890409148673372146500904E-2,
-1.487073146698099509605046333E-3,
-1.38040145414143859607708920E-4,
-1.1280303332287491498507366E-5,
-1.172869842743725224053739E-6,
-1.03476150393304615537382E-7,
-1.1899114085892438254447E-8,
-1.016222544989498640476E-9,
-1.37895716146965692169E-10,
-9.369613033737303335E-12,
-1.918809583959525349E-12,
-3.7573017201993707E-14,
-3.7053726026983357E-14,
2.627565423490371E-15,
-1.121322876437933E-15,
1.84136028922538E-16,
-4.9130256574886E-17,
1.0704455167373E-17,
-2.671893662405E-18,
6.49326867976E-19,
-1.65399353183E-19,
4.2605626604E-20,
-1.1255840765E-20,
3.025617448E-21,
-8.29042146E-22,
2.31049558E-22,
-6.5469511E-23,
1.8842314E-23,
-5.504341E-24,
1.630950E-24,
-4.89860E-25,
1.49054E-25,
-4.5922E-26,
1.4318E-26,
-4.516E-27,
1.440E-27,
-4.64E-28,
1.51E-28,
-5.0E-29,
1.7E-29,
-6.0E-30,
2.0E-30,
-1.0E-30
};
double DE1[] =
{
-0.689379974848418501361491576718,
0.295939056851161774752959335568,
-0.087237828075228616420029484096,
0.019959734091835509766546612696,
-0.003740200486895490324750329974,
0.000593337912367800463413186784,
-0.000081560801047403878256504204,
9.886099179971884018535968E-6,
-1.071209234904290565745194E-6,
1.0490945447626050322784E-7,
-9.370959271038746709966E-9,
7.6927263488753841874E-10,
-5.8412335114551520146E-11,
4.125393291736424788E-12,
-2.72304624901729048E-13,
1.6869717361387012E-14,
-9.84565340276638E-16,
5.4313471880068E-17,
-2.840458699772E-18,
1.4120512798E-19,
-6.688772574E-21,
3.0257558E-22,
-1.3097526E-23,
5.4352E-25,
-2.1704E-26,
8.32E-28,
-5.4E-29
};
double DE2[] =
{
0.717710208167480928473053690384,
-0.379868973985143305103199928808,
0.125832094465157378967135019248,
-0.030917661684228839423081992424,
0.006073689914144320367855343072,
-0.000996057789064916825079352632,
0.000140310790466315733723475232,
-0.000017328176496070286001302184,
1.90540194670935746397168e-6,
-1.8882873760163694937908e-7,
1.703176613666840587056e-8,
-1.40955218086201517976e-9,
1.0776816914256065828e-10,
-7.656138112778696256e-12,
5.07943557413613792e-13,
-3.1608615530282912e-14,
1.852036572003432e-15,
-1.02524641430496e-16,
5.37852808112e-18,
-2.68128238704e-19,
1.273321788e-20,
-5.77335744e-22,
2.504352e-23,
-1.0446e-24,
4.16e-26,
-2.808e-27
};
// inline functions for performance
/* ---------------------------------------------------------------------- */
inline double ipoly02(double x)
{
/* P(x) in the range x > 2 */
int i;
double b0, b1, b2;
b1 = 0.0; b0 = 0.0; x *= 2;
for (i = ERF_TERMS2; i >= 0; i--)
{
b2 = b1;
b1 = b0;
b0 = x * b1 - b2 + E2[i];
}
return 0.5 * (b0 - b2);
}
/* ---------------------------------------------------------------------- */
inline double ipoly1(double x)
{
/* First derivative P'(x) in the range x < 2 */
int i;
double b0, b1, b2;
b1 = 0.0; b0 = 0.0; x *= 2;
for (i = DERF_TERMS; i >= 0; i--)
{
b2 = b1;
b1 = b0;
b0 = x * b1 - b2 + DE1[i];
}
return 0.5 * (b0 - b2);
}
/* ---------------------------------------------------------------------- */
inline double ipoly01(double x)
{
// P(x) in the range x < 2
int i;
double b0, b1, b2;
b1 = 0.0; b0 = 0.0; x *= 2;
for (i = ERF_TERMS1; i >= 0; i--)
{
b2 = b1;
b1 = b0;
b0 = x * b1 - b2 + E1[i];
}
return 0.5 * (b0 - b2);
}
/* ---------------------------------------------------------------------- */
inline double ierfoverx1(double x, double *df)
{
// Computes Erf(x)/x and its first derivative
double t, f;
double x2; // x squared
double exp_term, recip_x;
if (x < 2.0)
{
/* erf(x) = x * y(t) */
/* t = 2 * (x/2)^2 - 1. */
t = 0.5 * x * x - 1;
f = ipoly01(t);
*df = ipoly1(t) * x;
}
else
{
/* erf(x) = 1 - exp(-x^2)/x * y(t) */
/* t = (10.5 - x^2) / (2.5 + x^2) */
x2 = x * x;
t = (10.5 - x2) / (2.5 + x2);
exp_term = exp(-x2);
recip_x = 1.0 / x;
f = 1.0 / x - (exp_term / x2) * ipoly02(t);
*df = (1.12837916709551257389615890312 * exp_term - f) * recip_x;
}
return f;
}
/* ---------------------------------------------------------------------- */
inline void KinElec(double radius, double *eke, double *frc)
{
*eke += 1.5 / (radius * radius);
*frc += 3.0 / (radius * radius * radius);
}
/* ---------------------------------------------------------------------- */
inline void ElecNucNuc(double q, double rc, double *ecoul, double *frc)
{
*ecoul += q / rc;
*frc += q / (rc * rc);
}
/* ---------------------------------------------------------------------- */
inline void ElecNucElec(double q, double rc, double re1,
double *ecoul, double *frc, double *fre1)
{
double a, arc;
double coeff_a;
/* sqrt(2) */
coeff_a = 1.4142135623730951;
/* E = -Z/r Erf(a r / re) */
/* constants: sqrt(2), 2 / sqrt(pi) */
a = coeff_a / re1;
arc = a * rc;
/* Interaction between nuclear point charge and Gaussian electron */
double E, dEdr, dEdr1, f, df;
f = ierfoverx1(arc, &df);
dEdr = q * a * a * df;
dEdr1 = -q * (a / re1) * (f + arc * df);
E = -q * a * f;
*ecoul += E;
*frc += dEdr;
*fre1 += dEdr1;
}
/* ---------------------------------------------------------------------- */
inline void ElecElecElec(double rc, double re1, double re2,
double *ecoul, double *frc, double *fre1,
double *fre2)
{
double a, arc, re, fre;
double coeff_a;
/* sqrt(2) */
coeff_a = 1.4142135623730951;
re = sqrt(re1 * re1 + re2 * re2);
/* constants: sqrt(2), 2 / sqrt(pi) */
a = coeff_a / re;
arc = a * rc;
/* V_elecelec = E * F */
/* where E = -Z/r Erf(a r / re) */
/* F = (1 - (b s + c s^2) exp(-d s^2)) */
/* and s = r / re */
double E, dEdr, dEdr1, dEdr2, f, df;
f = ierfoverx1(arc, &df);
dEdr = -a * a * df;
fre = a * (f + arc * df) / (re * re);
dEdr1 = fre * re1;
dEdr2 = fre * re2;
E = a * f;
*ecoul += E;
*frc += dEdr;
*fre1 += dEdr1;
*fre2 += dEdr2;
}
/* ---------------------------------------------------------------------- */
inline void ElecCoreNuc(double q, double rc, double re1, double *ecoul, double *frc)
{
double a, arc;
double coeff_a;
double E, dEdr, df, f;
coeff_a = 1.4142135623730951; /* sqrt(2) */
a = coeff_a / re1;
arc = a * rc;
f = ierfoverx1(arc, &df);
dEdr = -q * a * a * df;
E = q * a * f;
*ecoul += E;
*frc += dEdr;
}
/* ---------------------------------------------------------------------- */
inline void ElecCoreCore(double q, double rc, double re1, double re2,
double *ecoul, double *frc)
{
double a, arc, re;
double coeff_a;
double E, dEdr, f, df;
coeff_a = 1.4142135623730951;
re = sqrt(re1 * re1 + re2 * re2);
a = coeff_a / re;
arc = a * rc;
f = ierfoverx1(arc, &df);
dEdr = -q * a * a * df;
E = q * a * f;
*ecoul += E;
*frc += dEdr;
}
/* ---------------------------------------------------------------------- */
inline void ElecCoreElec(double q, double rc, double re1, double re2,
double *ecoul, double *frc, double *fre2)
{
double a,arc, re;
double coeff_a;
double E, dEdr, dEdr2, f, df, fre;
/* sqrt(2) */
coeff_a = 1.4142135623730951;
/*
re1: core size
re2: electron size
re3: size of the core, obtained from the electron density function rho(r) of core
e.g. rho(r) = a1*exp(-((r)/b1)^2), a1 =157.9, b1 = 0.1441 -> re3 = 0.1441 for Si4+
*/
re = sqrt(re1 * re1 + re2 * re2);
a = coeff_a / re;
arc = a * rc;
f = ierfoverx1(arc, &df);
E = -q * a * f;
dEdr = -q * a * df * a;
fre = q * a * (f + arc * df) / (re * re);
dEdr2 = fre * re2;
*ecoul += E;
*frc -= dEdr;
*fre2 -= dEdr2;
}
/* ---------------------------------------------------------------------- */
inline void PauliElecElec(int samespin, double rc,
double re1, double re2, double *epauli,
double *frc, double *fre1, double *fre2)
{
double ree, rem;
double S, t1, t2, tt;
double dSdr1, dSdr2, dSdr;
double dTdr1, dTdr2, dTdr;
double O, dOdS, ratio;
re1 *= PAULI_RE; re2 *= PAULI_RE; rc *= PAULI_RC;
ree = re1 * re1 + re2 * re2;
rem = re1 * re1 - re2 * re2;
S = (2.82842712474619 / pow((re2 / re1 + re1 / re2), 1.5)) *
exp(-rc * rc / ree);
t1 = 1.5 * (1 / (re1 * re1) + 1 / (re2 * re2));
t2 = 2.0 * (3 * ree - 2 * rc * rc) / (ree * ree);
tt = t1 - t2;
dSdr1 = (-1.5 / re1) * (rem / ree) + 2 * re1 * rc * rc / (ree * ree);
dSdr2 = (1.5 / re2) * (rem / ree) + 2 * re2 * rc * rc / (ree * ree);
dSdr = -2 * rc / ree;
dTdr1 = -3 / (re1 * re1 * re1) - 12 * re1 / (ree * ree) + 8 * re1 *
(-2 * rc * rc + 3 * ree) / (ree * ree * ree);
dTdr2 = -3 / (re2 * re2 * re2) - 12 * re2 / (ree * ree) + 8 * re2 *
(-2 * rc * rc + 3 * ree) / (ree * ree * ree);
dTdr = 8 * rc / (ree * ree);
if (samespin == 1) {
O = S * S / (1.0 - S * S) + (1 - PAULI_RHO) * S * S / (1.0 + S * S);
dOdS = 2 * S / ((1.0 - S * S) * (1.0 - S * S)) +
(1 - PAULI_RHO) * 2 * S / ((1.0 + S * S) * (1.0 + S * S));
} else {
O = -PAULI_RHO * S * S / (1.0 + S * S);
dOdS = -PAULI_RHO * 2 * S / ((1.0 + S * S) * (1.0 + S * S));
}
ratio = tt * dOdS * S;
*fre1 -= PAULI_RE * (dTdr1 * O + ratio * dSdr1);
*fre2 -= PAULI_RE * (dTdr2 * O + ratio * dSdr2);
*frc -= PAULI_RC * (dTdr * O + ratio * dSdr);
*epauli += tt*O;
}
/* ---------------------------------------------------------------------- */
inline void PauliCoreElec(double rc, double re2, double *epauli, double *frc,
double *fre2, double PAULI_CORE_A, double PAULI_CORE_B, double PAULI_CORE_C)
{
double E, dEdrc, dEdre2, rcsq, ssq;
rcsq = rc * rc;
ssq = re2 * re2;
E = PAULI_CORE_A * exp((-PAULI_CORE_B * rcsq) / (ssq + PAULI_CORE_C));
dEdrc = -2 * PAULI_CORE_A * PAULI_CORE_B * rc * exp(-PAULI_CORE_B * rcsq /
(ssq + PAULI_CORE_C)) / (ssq + PAULI_CORE_C);
dEdre2 = 2 * PAULI_CORE_A * PAULI_CORE_B * re2 * rcsq * exp(-PAULI_CORE_B * rcsq /
(ssq + PAULI_CORE_C)) / ((PAULI_CORE_C + ssq)*(PAULI_CORE_C + ssq));
*epauli += E;
*frc -= dEdrc;
*fre2 -= dEdre2;
}
/* ---------------------------------------------------------------------- */
inline void PauliCorePElec(double rc, double re2, double *epauli, double *frc,
double *fre2, double PAULI_CORE_P_A,
double PAULI_CORE_P_B, double PAULI_CORE_P_C,
double PAULI_CORE_P_D, double PAULI_CORE_P_E)
{
double E, dEdrc, dEdre2;
E = PAULI_CORE_P_A *
pow((2.0 / (PAULI_CORE_P_B / re2 + re2 / PAULI_CORE_P_B)), 5.0) *
pow((rc - PAULI_CORE_P_C * re2), 2.0) * exp(-PAULI_CORE_P_D *
pow((rc - PAULI_CORE_P_C * re2), 2.0) / (PAULI_CORE_P_E + re2 * re2));
dEdrc = PAULI_CORE_P_A *
pow((2.0 / (PAULI_CORE_P_B / re2 + re2 / PAULI_CORE_P_B)), 5.0) *
2.0 * (rc - PAULI_CORE_P_C * re2) * exp(-PAULI_CORE_P_D *
pow((rc - PAULI_CORE_P_C * re2), 2.0) / (PAULI_CORE_P_E + re2 * re2))
+ E * (-PAULI_CORE_P_D * 2.0 * (rc - PAULI_CORE_P_C * re2) /
(PAULI_CORE_P_E + re2 * re2));
dEdre2 = E * (-5.0 / (PAULI_CORE_P_B / re2 + re2 / PAULI_CORE_P_B) *
(-PAULI_CORE_P_B / (re2 * re2) + 1.0 / PAULI_CORE_P_B))
+ PAULI_CORE_P_A *
pow((2.0 / (PAULI_CORE_P_B / re2 + re2 / PAULI_CORE_P_B)), 5.0) *
2.0 * (rc - PAULI_CORE_P_C * re2) * (-PAULI_CORE_P_C) *
exp(-PAULI_CORE_P_D * pow((rc - PAULI_CORE_P_C * re2), 2.0) /
(PAULI_CORE_P_E + re2 * re2)) + E * (2.0 * PAULI_CORE_P_D *
(rc - PAULI_CORE_P_C * re2) * (PAULI_CORE_P_C * PAULI_CORE_P_E +
rc * re2) / pow((PAULI_CORE_P_E + re2 * re2), 2.0));
*epauli += E;
*frc -= dEdrc;
*fre2 -= dEdre2;
}
/* ---------------------------------------------------------------------- */
inline void RForce(double dx, double dy, double dz,
double rc, double force, double *fx, double *fy, double *fz)
{
force /= rc;
*fx = force * dx;
*fy = force * dy;
*fz = force * dz;
}
/* ---------------------------------------------------------------------- */
inline void SmallRForce(double dx, double dy, double dz,
double rc, double force,
double *fx, double *fy, double *fz)
{
/* Handles case where rc is small to avoid division by zero */
if (rc > 0.000001){
force /= rc;
*fx = force * dx; *fy = force * dy; *fz = force * dz;
} else {
if (dx != 0) *fx = force / sqrt(1 + (dy * dy + dz * dz) / (dx * dx));
else *fx = 0.0;
if (dy != 0) *fy = force / sqrt(1 + (dx * dx + dz * dz) / (dy * dy));
else *fy = 0.0;
if (dz != 0) *fz = force / sqrt(1 + (dx * dx + dy * dy) / (dz * dz));
else *fz = 0.0;
// if (dx < 0) *fx = -*fx;
// if (dy < 0) *fy = -*fy;
// if (dz < 0) *fz = -*fz;
}
}
/* ---------------------------------------------------------------------- */
inline double cutoff(double x)
{
/* cubic: return x * x * (2.0 * x - 3.0) + 1.0; */
/* quintic: return -6 * pow(x, 5) + 15 * pow(x, 4) - 10 * pow(x, 3) + 1; */
/* Seventh order spline */
// return 20 * pow(x, 7) - 70 * pow(x, 6) + 84 * pow(x, 5) - 35 * pow(x, 4) + 1;
return (((20 * x - 70) * x + 84) * x - 35) * x * x * x * x + 1;
}
/* ---------------------------------------------------------------------- */
inline double dcutoff(double x)
{
/* cubic: return (6.0 * x * x - 6.0 * x); */
/* quintic: return -30 * pow(x, 4) + 60 * pow(x, 3) - 30 * pow(x, 2); */
/* Seventh order spline */
// return 140 * pow(x, 6) - 420 * pow(x, 5) + 420 * pow(x, 4) - 140 * pow(x, 3);
return (((140 * x - 420) * x + 420) * x - 140) * x * x * x;
}
}
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