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/*************************************************************************** |
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* This program is free software; you can redistribute it and/or modify * |
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* it under the terms of the GNU General Public License as published by * |
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* the Free Software Foundation; either version 3 of the License, or * |
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* (at your option) any later version. * |
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* * |
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* This program is distributed in the hope that it will be useful, * |
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* but WITHOUT ANY WARRANTY; without even the implied warranty of * |
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * |
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* GNU General Public License for more details. * |
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* * |
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* You should have received a copy of the GNU General Public License * |
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* along with this program; if not, see <http://www.gnu.org/licenses/>. * |
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***************************************************************************/ |
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|
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/*** |
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* This file was imported from qtpie, found at http://code.google.com/p/qtpie |
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* and was modified minimally for use in OpenMD. |
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* |
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* No author attribution was found in the code, but it presumably is |
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* the work of J. Chen and Todd J. Martinez. |
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* |
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* QTPIE (charge transfer with polarization current equilibration) is |
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* a new charge model, similar to other charge models like QEq, |
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* fluc-q, EEM or ABEEM. Unlike other existing charge models, however, |
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* it is capable of describing both charge transfer and polarization |
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* phenomena. It is also unique for its ability to describe |
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* intermolecular charge transfer at reasonable computational cost. |
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* |
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* Good references to cite when using this code are: |
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* |
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* J. Chen and T. J. Martinez, "QTPIE: Charge transfer with |
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* polarization current equalization. A fluctuating charge model with |
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* correct asymptotics", Chemical Physics Letters 438 (2007), |
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* 315-320. DOI: 10.1016/j.cplett.2007.02.065. Erratum: ibid, 463 |
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* (2008), 288. DOI: 10.1016/j.cplett.2008.08.060 |
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* |
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* J. Chen, D. Hundertmark and T. J. Martinez, "A unified theoretical |
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* framework for fluctuating-charge models in atom-space and in |
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* bond-space", Journal of Chemical Physics 129 (2008), 214113. DOI: |
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* 10.1063/1.3021400. |
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* |
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* J. Chen and T. J. Martinez, "Charge conservation in |
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* electronegativity equalization and its implications for the |
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* electrostatic properties of fluctuating-charge models", Journal of |
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* Chemical Physics 131 (2009), 044114. DOI: 10.1063/1.3183167 |
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* |
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* J. Chen and T. J. Martinez, "The dissociation catastrophe in |
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* fluctuating-charge models and its implications for the concept of |
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* atomic electronegativity", Progress in Theoretical Chemistry and |
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* Physics, to appear. arXiv:0812.1543 |
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* |
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* J. Chen, "Theory and applications of fluctuating-charge models", |
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* PhD (chemical physics) thesis, University of Illinois at |
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* Urbana-Champaign, Department of Chemistry, 2009. |
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* |
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* J. Chen and T. J. Martinez, "Size-extensive polarizabilities with |
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* intermolecular charge transfer in a fluctuating-charge model", in |
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* preparation. arXiv:0812.1544 |
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*/ |
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|
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#include "config.h" |
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#include <cmath> |
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#include <cstdlib> |
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#include <iostream> |
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#include "math/Factorials.hpp" |
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|
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#ifndef NONBONDED_SLATERINTEGRALS_HPP |
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#define NONBONDED_SLATERINTEGRALS_HPP |
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|
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template <typename T> inline T sqr(T t) { return t*t; } |
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template <typename T> inline T mod(T x, T m) |
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{ return x<0 ? m - 1 - ((-x) - 1)%m : x%m; } |
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|
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// #include "parameters.h" |
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|
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/** |
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* @brief Computes Rosen's Guillimer-Zener function A |
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* Computes Rosen's A integral, an auxiliary quantity needed to |
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* compute integrals involving Slater-type orbitals of s symmetry. |
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* \f[ |
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* A_n(\alpha) = \int_1^\infty x^n e^{-\alpha x}dx |
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* = \frac{n! e^{-\alpha}}{\alpha^{n+1}}\sum_{\nu=0}^n |
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* \frac{\alpha^\nu}{\nu!} |
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* \f] |
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* @param n - principal quantum number |
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* @param alpha - Slater exponent |
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* @return the value of Rosen's A integral |
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* @note N. Rosen, Phys. Rev., 38 (1931), 255 |
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*/ |
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inline RealType RosenA(int n, RealType a) |
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{ |
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RealType RosenA_ = 0.; |
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if (a != 0.) |
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{ |
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RealType Term = 1.; |
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RosenA_ = Term; |
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for (int nu=1; nu<=n; nu++) |
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{ |
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Term *= a / nu; |
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RosenA_ += Term; |
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} |
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RosenA_ = (RosenA_/Term) * (exp(-a)/a); |
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} |
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return RosenA_; |
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} |
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|
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/** |
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* @brief Computes Rosen's Guillimer-Zener function B |
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* Computes Rosen's B integral, an auxiliary quantity needed to |
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* compute integrals involving Slater-type orbitals of s symmetry. |
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* \f[ |
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* B_n(\alpha) = \int_{-1}^1 x^n e^{-\alpha x} dx |
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* = \frac{n!}{\alpha^{n+1}} |
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* \sum_{\nu=0}^n \frac{e^\alpha(-\alpha)^\nu |
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* - e^{-\alpha} \alpha^\nu}{\nu!} |
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* \f] |
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* @param n - principal quantum number |
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* @param alpha - Slater exponent |
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* @return the value of Rosen's B integral |
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* @note N. Rosen, Phys. Rev., 38 (1931), 255 |
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*/ |
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inline RealType RosenB(int n, RealType alpha) |
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{ |
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RealType TheSum, Term; |
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RealType RosenB_, PSinhRosenA, PCoshRosenA, PHyperRosenA; |
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int nu; |
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bool IsPositive; |
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if (alpha != 0.) |
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{ |
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Term = 1.; |
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TheSum = 1.; |
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IsPositive = true; |
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|
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// These two expressions are (up to constant factors) equivalent |
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// to computing the hyperbolic sine and cosine of a respectively |
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// The series consists of adding up these terms in an alternating fashion |
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PSinhRosenA = exp(alpha) - exp(-alpha); |
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PCoshRosenA = -exp(alpha) - exp(-alpha); |
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TheSum = PSinhRosenA; |
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for (unsigned nu=1; nu<=n; nu++) |
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{ |
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if (IsPositive) |
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{ |
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PHyperRosenA = PCoshRosenA; |
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IsPositive = false; |
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} |
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else // term to add should be negative |
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{ |
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PHyperRosenA = PSinhRosenA; |
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IsPositive = true; |
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} |
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Term *= alpha / nu; |
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TheSum += Term * PHyperRosenA; |
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} |
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RosenB_ = TheSum / (alpha*Term); |
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} |
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else // pathological case of a=0 |
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{ |
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printf("WARNING, a = 0 in RosenB\n"); |
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RosenB_ = (1. - pow(-1., n)) / (n + 1.); |
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} |
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return RosenB_; |
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} |
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|
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/** @brief Computes Rosen's D combinatorial factor |
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* Computes Rosen's D factor, an auxiliary quantity needed to |
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* compute integrals involving Slater-type orbitals of s symmetry. |
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* \f[ |
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* RosenD^{mn}_p = \sum_k (-1)^k \frac{m! n!} |
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* {(p-k)!(m-p+k)!(n-k)!k!} |
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* \f] |
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* @return the value of Rosen's D factor |
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* @note N. Rosen, Phys. Rev., 38 (1931), 255 |
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*/ |
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inline RealType RosenD(int m, int n, int p) |
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{ |
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if (m+n+p > maxFact) |
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{ |
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printf("Error, arguments exceed maximum factorial computed %d > %d\n", m+n+p, maxFact); |
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::exit(0); |
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} |
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|
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RealType RosenD_ = 0; |
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for (int k=max(p-m,0); k<=min(n,p); k++) |
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{ |
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if (mod(k,2) == 0) |
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RosenD_ += (fact[m] / (fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k])); |
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else |
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RosenD_ -= (fact[m] / ( fact[p-k] * fact[m-p+k])) * (fact[n] / (fact[n-k] * fact[k])); |
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} |
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return RosenD_; |
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} |
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|
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/** @brief Computes Coulomb integral analytically over s-type STOs |
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* Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry. |
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* @param a : Slater zeta exponent of first atom in inverse Bohr (au) |
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* @param b : Slater zeta exponent of second atom in inverse Bohr (au) |
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* @param m : principal quantum number of first atom |
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* @param n : principal quantum number of second atom |
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* @param R : internuclear distance in atomic units (bohr) |
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* @return value of the Coulomb potential energy integral |
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* @note N. Rosen, Phys. Rev., 38 (1931), 255 |
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* @note In Rosen's paper, this integral is known as K2. |
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*/ |
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inline RealType sSTOCoulInt(RealType a, RealType b, int m, int n, RealType R) |
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{ |
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int nu, p; |
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RealType x, K2; |
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RealType Factor1, Factor2, Term, OneElectronTerm; |
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RealType eps, epsi; |
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|
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// To speed up calculation, we terminate loop once contributions |
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// to integral fall below the bound, epsilon |
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RealType epsilon = 0.; |
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|
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// x is the argument of the auxiliary RosenA and RosenB functions |
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x = 2. * a * R; |
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|
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// First compute the two-electron component |
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RealType sSTOCoulInt_ = 0.; |
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if (x == 0.) // Pathological case |
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{ |
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if ((a==b) && (m==n)) |
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{ |
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for (int nu=0; nu<=2*n-1; nu++) |
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{ |
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K2 = 0.; |
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for (unsigned p=0; p<=2*n+m; p++) K2 += 1. / fact[p]; |
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sSTOCoulInt_ += K2 * fact[2*n+m] / fact[m]; |
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} |
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sSTOCoulInt_ = 2 * a / (n * fact[2*n]) * sSTOCoulInt_; |
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} |
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else |
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{ |
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// Not implemented |
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cerr << "ERROR, sSTOCoulInt cannot compute from arguments\n"; |
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cerr << "a = " << a << "\tb = " << b << "\tm = " << m <<"\tn = " << n << "\t R = " << R << "\n"; |
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//printf("ERROR, sSTOCoulInt cannot compute from arguments\n"); |
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//printf("a = %lf b = %lf m = %d n = %d R = %lf\n",a, b, m, n, R); |
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//exit(0); |
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} |
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} |
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else |
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{ |
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OneElectronTerm = 1./R + pow(x, 2*m)/(fact[2*m]*R)* |
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((x-2*m)*RosenA(2*m-1,x)-exp(-x)) + sSTOCoulInt_; |
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eps = epsilon / OneElectronTerm; |
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if (a == b) |
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{ |
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// Apply Rosen (48) |
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Factor1 = -a*pow(a*R, 2*m)/(n*fact[2*m]); |
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for (int nu=0; nu<=2*n-1; nu++) |
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{ |
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Factor2 = (2.*n-nu)/fact[nu]*pow(a*R,nu); |
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epsi = eps / fabs(Factor1 * Factor2); |
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K2 = 0.; |
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for (int p=0; p<=m+(nu-1)/2; p++) |
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{ |
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Term = RosenD(2*m-1, nu, 2*p)/(2.*p+1.) *RosenA(2*m+nu-1-2*p,x); |
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K2 += Term; |
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if ((Term > 0) && (Term < epsi)) goto label1; |
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} |
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sSTOCoulInt_ += K2 * Factor2; |
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} |
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label1: |
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sSTOCoulInt_ *= Factor1; |
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} |
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else |
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{ |
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Factor1 = -a*pow(a*R,2*m)/(2.*n*fact[2*m]); |
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epsi = eps/fabs(Factor1); |
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if (b == 0.) |
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printf("WARNING: b = 0 in sSTOCoulInt\n"); |
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else |
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{ |
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// Apply Rosen (54) |
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for (int nu=0; nu<=2*n-1; nu++) |
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{ |
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K2 = 0; |
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for (int p=0; p<=2*m+nu-1; p++) |
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K2=K2+RosenD(2*m-1,nu,p)*RosenB(p,R*(a-b)) |
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*RosenA(2*m+nu-1-p,R*(a+b)); |
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Term = K2*(2*n-nu)/fact[nu]*pow(b*R, nu); |
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sSTOCoulInt_ += Term; |
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if (fabs(Term) < epsi) goto label2; |
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} |
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label2: |
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sSTOCoulInt_ *= Factor1; |
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} |
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} |
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// Now add the one-electron term from Rosen (47) = Rosen (53) |
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sSTOCoulInt_ += OneElectronTerm; |
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} |
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return sSTOCoulInt_; |
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} |
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|
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/** |
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* @brief Computes overlap integral analytically over s-type STOs |
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* Computes the overlap integral over two |
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* Slater-type orbitals of s symmetry. |
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* @param a : Slater zeta exponent of first atom in inverse Bohr (au) |
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* @param b : Slater zeta exponent of second atom in inverse Bohr (au) |
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* @param m : principal quantum number of first atom |
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* @param n : principal quantum number of second atom |
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* @param R : internuclear distance in atomic units (bohr) |
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* @return the value of the sSTOOvInt integral |
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* @note N. Rosen, Phys. Rev., 38 (1931), 255 |
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* @note In the Rosen paper, this integral is known as I. |
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*/ |
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inline RealType sSTOOvInt(RealType a, RealType b, int m, int n, RealType R) |
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{ |
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RealType Factor, Term, eps; |
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|
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// To speed up calculation, we terminate loop once contributions |
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// to integral fall below the bound, epsilon |
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RealType epsilon = 0.; |
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RealType sSTOOvInt_ = 0.; |
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|
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if (a == b) |
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{ |
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Factor = pow(a*R, m+n+1)/sqrt(fact[2*m]*fact[2*n]); |
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eps = epsilon / fabs(Factor); |
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for (int q=0; q<=(m+n)/2; q++) |
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{ |
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Term = RosenD(m,n,2*q)/(2.*q+1.)*RosenA(m+n-2*q,a*R); |
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sSTOOvInt_ += Term; |
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if (fabs(Term) < eps) exit(0); |
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} |
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sSTOOvInt_ *= Factor; |
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} |
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else |
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{ |
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Factor = 0.5*pow(a*R, m+0.5)*pow(b*R,n+0.5) |
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/sqrt(fact[2*m]*fact[2*n]); |
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eps = epsilon / fabs(Factor); |
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for (int q=0; q<=m+n; q++) |
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{ |
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Term = RosenD(m,n,q)*RosenB(q, R/2.*(a-b)) |
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* RosenA(m+n-q,R/2.*(a+b)); |
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sSTOOvInt_ += Term; |
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if (fabs(Term) < eps) exit(0); |
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} |
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sSTOOvInt_ *= Factor; |
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} |
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return sSTOOvInt_; |
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} |
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|
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/** |
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* @brief Computes kinetic energy integral analytically over s-type STOs |
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* Computes the overlap integral over two Slater-type orbitals of s symmetry. |
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* @param a : Slater zeta exponent of first atom in inverse Bohr (au) |
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* @param b : Slater zeta exponent of second atom in inverse Bohr (au) |
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* @param m : principal quantum number of first atom |
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* @param n : principal quantum number of second atom |
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* @param R : internuclear distance in atomic units (bohr) |
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* @return the value of the kinetic energy integral |
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* @note N. Rosen, Phys. Rev., 38 (1931), 255 |
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* @note untested |
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*/ |
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inline RealType KinInt(RealType a, RealType b, int m, int n,RealType R) |
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{ |
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RealType KinInt_ = -0.5*b*b*sSTOOvInt(a, b, m, n, R); |
| 364 |
if (n > 0) |
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{ |
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KinInt_ += b*b*pow(2*b/(2*b-1),0.5) * sSTOOvInt(a, b, m, n-1, R); |
| 367 |
if (n > 1) KinInt_ += pow(n*(n-1)/((n-0.5)*(n-1.5)), 0.5) |
| 368 |
* sSTOOvInt(a, b, m, n-2, R); |
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} |
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return KinInt_; |
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} |
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|
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/** |
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* @brief Computes derivative of Coulomb integral with respect to the interatomic distance |
| 375 |
* Computes the two-center Coulomb integral over Slater-type orbitals of s symmetry. |
| 376 |
* @param a: Slater zeta exponent of first atom in inverse Bohr (au) |
| 377 |
* @param b: Slater zeta exponent of second atom in inverse Bohr (au) |
| 378 |
* @param m: principal quantum number of first atom |
| 379 |
* @param n: principal quantum number of second atom |
| 380 |
* @param R: internuclear distance in atomic units (bohr) |
| 381 |
* @return the derivative of the Coulomb potential energy integral |
| 382 |
* @note Derived in QTPIE research notes, May 15 2007 |
| 383 |
*/ |
| 384 |
inline RealType sSTOCoulIntGrad(RealType a, RealType b, int m, int n, RealType R) |
| 385 |
{ |
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RealType x, y, z, K2, TheSum; |
| 387 |
// x is the argument of the auxiliary RosenA and RosenB functions |
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x = 2. * a * R; |
| 389 |
|
| 390 |
// First compute the two-electron component |
| 391 |
RealType sSTOCoulIntGrad_ = 0.; |
| 392 |
if (x==0) // Pathological case |
| 393 |
{ |
| 394 |
printf("WARNING: argument given to sSTOCoulIntGrad is 0\n"); |
| 395 |
printf("a = %lf R= %lf\n", a, R); |
| 396 |
} |
| 397 |
else |
| 398 |
{ |
| 399 |
if (a == b) |
| 400 |
{ |
| 401 |
TheSum = 0.; |
| 402 |
for (int nu=0; nu<=2*(n-1); nu++) |
| 403 |
{ |
| 404 |
K2 = 0.; |
| 405 |
for (int p=0; p<=(m+nu)/2; p++) |
| 406 |
K2 += RosenD(2*m-1, nu+1, 2*p)/(2*p + 1.) * RosenA(2*m+nu-1-2*p, x); |
| 407 |
TheSum += (2*n-nu-1)/fact[nu]*pow(a*R, nu) * K2; |
| 408 |
} |
| 409 |
sSTOCoulIntGrad_ = -pow(a, 2*m+2)*pow(R, 2*m) /(n*fact[2*m])*TheSum; |
| 410 |
TheSum = 0.; |
| 411 |
for (int nu=0; nu<=2*n-1; nu++) |
| 412 |
{ |
| 413 |
K2 = 0.; |
| 414 |
for (int p=0; p<=(m+nu-1)/2; p++) |
| 415 |
K2 += RosenD(2*m-1, nu, 2*p)/(2*p + 1.) * RosenA(2*m+nu-2*p, x); |
| 416 |
TheSum += (2*n-nu)/fact[nu]*pow(a*R,nu) * K2; |
| 417 |
} |
| 418 |
sSTOCoulIntGrad_ += 2*pow(a, 2*m+2)*pow(R, 2*m) /(n*fact[2*m])*TheSum; |
| 419 |
} |
| 420 |
else |
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{ |
| 422 |
// Slater exponents are different |
| 423 |
// First calculate some useful arguments |
| 424 |
y = R*(a+b); |
| 425 |
z = R*(a-b); |
| 426 |
TheSum = 0.; |
| 427 |
for (int nu=0; nu<=2*n-1; nu++) |
| 428 |
{ |
| 429 |
K2 = 0.; |
| 430 |
for (int p=0; p<=2*m+nu; p++) |
| 431 |
K2 += RosenD(2*m-1, nu+1, p) |
| 432 |
* RosenB(p,z)*RosenA(2*m+nu-p, y); |
| 433 |
TheSum += (2*n-nu-1)/fact[nu]*pow(b*R,nu) * K2; |
| 434 |
} |
| 435 |
sSTOCoulIntGrad_ = -b*pow(a,2*m+1)*pow(R,2*m)/ |
| 436 |
(2*n*fact[2*m])*TheSum; |
| 437 |
TheSum = 0.; |
| 438 |
for (int nu=0; nu<=2*n; nu++) |
| 439 |
{ |
| 440 |
K2 = 0.; |
| 441 |
for (int p=0; p<=2*m-1+nu; p++) |
| 442 |
K2 += RosenD(2*m-1, nu, p) |
| 443 |
* ((a-b)*RosenB(p+1,z)*RosenA(2*m+nu-p-1, y) |
| 444 |
+(a+b)*RosenB(p ,z)*RosenA(2*m+nu-p , y)); |
| 445 |
TheSum += (2*n-nu)/fact[nu]*pow(b*R,nu) * K2; |
| 446 |
} |
| 447 |
sSTOCoulIntGrad_ += pow(a,2*m+1)*pow(R,2*m)/(2*n*fact[2*m])*TheSum; |
| 448 |
} |
| 449 |
// Now add one-electron terms and common term |
| 450 |
sSTOCoulIntGrad_ = sSTOCoulIntGrad_ - (2.*m+1.)/sqr(R) |
| 451 |
+ 2.*m/R * sSTOCoulInt(a,b,m,n,R) |
| 452 |
+ pow(x,2*m)/(fact[2*m]*sqr(R)) * ((2.*m+1.)*exp(-x) |
| 453 |
+ 2.*m*(1.+2.*m-x)*RosenA(2*m-1,x)); |
| 454 |
} |
| 455 |
return sSTOCoulIntGrad_; |
| 456 |
} |
| 457 |
|
| 458 |
/** |
| 459 |
* @brief Computes gradient of overlap integral with respect to the interatomic diatance |
| 460 |
* Computes the derivative of the overlap integral over two Slater-type orbitals of s symmetry. |
| 461 |
* @param a: Slater zeta exponent of first atom in inverse Bohr (au) |
| 462 |
* @param b: Slater zeta exponent of second atom in inverse Bohr (au) |
| 463 |
* @param m: principal quantum number of first atom |
| 464 |
* @param n: principal quantum number of second atom |
| 465 |
* @param R: internuclear distance in atomic units (bohr) |
| 466 |
* @return the derivative of the sSTOOvInt integral |
| 467 |
* @note Derived in QTPIE research notes, May 15 2007 |
| 468 |
*/ |
| 469 |
inline RealType sSTOOvIntGrad(RealType a, RealType b, int m, int n, RealType R) |
| 470 |
{ |
| 471 |
RealType w, x, y, z, TheSum; |
| 472 |
|
| 473 |
// Calculate first term |
| 474 |
RealType sSTOOvIntGrad_ = (m+n+1.)/R * sSTOOvInt(a, b, m, n, R); |
| 475 |
|
| 476 |
// Calculate remaining terms; answers depend on exponents |
| 477 |
TheSum = 0.; |
| 478 |
x = a * R; |
| 479 |
if (a == b) |
| 480 |
{ |
| 481 |
for (int q=0; q<=(m+n)/2; q++) |
| 482 |
TheSum += RosenD(m,n,2*q) / (2*q + 1.) * RosenA(m+n-2*q+1, x); |
| 483 |
sSTOOvIntGrad_ -= a*pow(x,m+n+1)/ sqrt(fact[2*m]*fact[2*n])*TheSum; |
| 484 |
} |
| 485 |
else |
| 486 |
{ |
| 487 |
w = b*R; |
| 488 |
y = 0.5*R*(a+b); |
| 489 |
z = 0.5*R*(a-b); |
| 490 |
for (int q=0; q<m+n; q++) |
| 491 |
TheSum = TheSum + RosenD(m,n,q) * |
| 492 |
((a-b)*RosenB(q+1,z)*RosenA(m+n-q ,y) |
| 493 |
+(a+b)*RosenB(q ,z)*RosenA(m+n-q+1,y)); |
| 494 |
sSTOOvIntGrad_ -= 0.25*sqrt((pow(x, 2*m+1)*pow(w, 2*n+1))/(fact[2*m]*fact[2*n]))*TheSum; |
| 495 |
} |
| 496 |
return sSTOOvIntGrad_; |
| 497 |
} |
| 498 |
|
| 499 |
/** |
| 500 |
* @brief Calculates a Slater-type orbital exponent based on the hardness parameters |
| 501 |
* @param Hardness: chemical hardness in atomic units |
| 502 |
* @param n: principal quantum number |
| 503 |
* @note Modified for use with OpenMD by Gezelter and Michalka. |
| 504 |
*/ |
| 505 |
inline RealType getSTOZeta(int n, RealType hardness) |
| 506 |
{ |
| 507 |
// Approximate the exact value of the constant of proportionality |
| 508 |
// by its value at a very small distance epsilon |
| 509 |
// since the exact R = 0 case has not be programmed |
| 510 |
RealType epsilon = 1.0e-8; |
| 511 |
|
| 512 |
// Assign orbital exponent |
| 513 |
return pow(sSTOCoulInt(1., 1., n, n, epsilon) / hardness, -1./(3. + 2.*n)); |
| 514 |
} |
| 515 |
|
| 516 |
#endif |
