/usr/include/chemps2/Cumulant.h is in libchemps2-dev 1.8.3-2.
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CheMPS2: a spin-adapted implementation of DMRG for ab initio quantum chemistry
Copyright (C) 2013-2016 Sebastian Wouters
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License along
with this program; if not, write to the Free Software Foundation, Inc.,
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*/
#ifndef CUMULANT_CHEMPS2_H
#define CUMULANT_CHEMPS2_H
#include "ThreeDM.h"
#include "TwoDM.h"
namespace CheMPS2{
/** Cumulant class.
\author Sebastian Wouters <sebastianwouters@gmail.com>
\date November 26, 2015
The cumulant class contains routines to approximate the spinfree 4-RDM \f$ \Gamma^4 \f$ by neglecting the fourth order cumulant \f$ \Lambda^4 \f$. Based on the spinfree density matrices
\f{eqnarray*}{
\Gamma^1_{ip} & = & \sum\limits_{\sigma} \braket{ 0 | \hat{a}_{i \sigma}^{\dagger} \hat{a}_{p \sigma} | 0 } \\
\Gamma^2_{ijpq} & = & \sum\limits_{\sigma \tau} \braket{ 0 | \hat{a}_{i \sigma}^{\dagger} \hat{a}_{j \tau}^{\dagger} \hat{a}_{q \tau} \hat{a}_{p \sigma} | 0 } \\
\Gamma^3_{ijkpqr} & = & \sum\limits_{\sigma \tau \chi} \braket{ 0 | \hat{a}_{i \sigma}^{\dagger} \hat{a}_{j \tau}^{\dagger} \hat{a}_{k \chi}^{\dagger} \hat{a}_{r \chi} \hat{a}_{q \tau} \hat{a}_{p \sigma} | 0 } \\
\Gamma^4_{ijklpqrs} & = & \sum\limits_{\sigma \tau \chi \omega } \braket{ 0 | \hat{a}_{i \sigma}^{\dagger} \hat{a}_{j \tau}^{\dagger} \hat{a}_{k \chi}^{\dagger} \hat{a}_{l \omega}^{\dagger} \hat{a}_{s \omega} \hat{a}_{r \chi} \hat{a}_{q \tau} \hat{a}_{p \sigma} | 0 }
\f}
and the spinfree second order cumulant
\f{eqnarray*}{
\Lambda^2_{ijpq} & = & \Gamma^2_{ijpq} - \Gamma^1_{ip} \Gamma^1_{jq} + \frac{1}{2} \Gamma^1_{iq} \Gamma^1_{jp}
\f}
the spinfree 4-RDM can be written as [CUM1]:
\f{eqnarray*}{
& & \Gamma^4_{ijklpqrs} \\
& = & \Lambda^4_{ijklpqrs} \\
& + & \Gamma^3_{ijkpqr} \Gamma^1_{ls}
- \frac{1}{2} \Gamma^3_{ijksqr} \Gamma^1_{lp}
- \frac{1}{2} \Gamma^3_{ijkpsr} \Gamma^1_{lq}
- \frac{1}{2} \Gamma^3_{ijkpqs} \Gamma^1_{lr} \\
& + & \Gamma^3_{ijlpqs} \Gamma^1_{kr}
- \frac{1}{2} \Gamma^3_{ijlrqs} \Gamma^1_{kp}
- \frac{1}{2} \Gamma^3_{ijlprs} \Gamma^1_{kq}
- \frac{1}{2} \Gamma^3_{ijlpqr} \Gamma^1_{ks} \\
& + & \Gamma^3_{iklprs} \Gamma^1_{jq}
- \frac{1}{2} \Gamma^3_{iklqrs} \Gamma^1_{jp}
- \frac{1}{2} \Gamma^3_{iklpqs} \Gamma^1_{jr}
- \frac{1}{2} \Gamma^3_{iklprq} \Gamma^1_{js} \\
& + & \Gamma^3_{jklqrs} \Gamma^1_{ip}
- \frac{1}{2} \Gamma^3_{jklprs} \Gamma^1_{iq}
- \frac{1}{2} \Gamma^3_{jklqps} \Gamma^1_{ir}
- \frac{1}{2} \Gamma^3_{jklqrp} \Gamma^1_{is} \\
& - & \Gamma^2_{ijpq} \Gamma^2_{klrs}
+ \frac{1}{2} \Gamma^2_{ijpr} \Gamma^2_{klqs}
+ \frac{1}{2} \Gamma^2_{ijps} \Gamma^2_{klrq}
+ \frac{1}{2} \Gamma^2_{ijrq} \Gamma^2_{klps}
+ \frac{1}{2} \Gamma^2_{ijsq} \Gamma^2_{klrp} \\
& - & \frac{1}{3} \Gamma^2_{ijrs} \Gamma^2_{klpq}
- \frac{1}{6} \Gamma^2_{ijrs} \Gamma^2_{klqp}
- \frac{1}{6} \Gamma^2_{ijsr} \Gamma^2_{klpq}
- \frac{1}{3} \Gamma^2_{ijsr} \Gamma^2_{klqp} \\
& - & \Gamma^2_{ikpr} \Gamma^2_{jlqs}
+ \frac{1}{2} \Gamma^2_{ikpq} \Gamma^2_{jlrs}
+ \frac{1}{2} \Gamma^2_{ikps} \Gamma^2_{jlqr}
+ \frac{1}{2} \Gamma^2_{ikqr} \Gamma^2_{jlps}
+ \frac{1}{2} \Gamma^2_{iksr} \Gamma^2_{jlqp} \\
& - & \frac{1}{3} \Gamma^2_{ikqs} \Gamma^2_{jlpr}
- \frac{1}{6} \Gamma^2_{iksq} \Gamma^2_{jlpr}
- \frac{1}{6} \Gamma^2_{ikqs} \Gamma^2_{jlrp}
- \frac{1}{3} \Gamma^2_{iksq} \Gamma^2_{jlrp} \\
& - & \Gamma^2_{ilps} \Gamma^2_{kjrq}
+ \frac{1}{2} \Gamma^2_{ilpr} \Gamma^2_{kjsq}
+ \frac{1}{2} \Gamma^2_{ilpq} \Gamma^2_{kjrs}
+ \frac{1}{2} \Gamma^2_{ilrs} \Gamma^2_{kjpq}
+ \frac{1}{2} \Gamma^2_{ilqs} \Gamma^2_{kjrp} \\
& - & \frac{1}{3} \Gamma^2_{ilrq} \Gamma^2_{kjps}
- \frac{1}{6} \Gamma^2_{ilqr} \Gamma^2_{kjps}
- \frac{1}{6} \Gamma^2_{ilrq} \Gamma^2_{kjsp}
- \frac{1}{3} \Gamma^2_{ilqr} \Gamma^2_{kjsp} \\
& + & 2 \Lambda^2_{ijpq} \Lambda^2_{klrs}
- \Lambda^2_{ijpr} \Lambda^2_{klqs}
- \Lambda^2_{ijps} \Lambda^2_{klrq}
- \Lambda^2_{ijrq} \Lambda^2_{klps}
- \Lambda^2_{ijsq} \Lambda^2_{klrp} \\
& + & \frac{2}{3} \Lambda^2_{ijrs} \Lambda^2_{klpq}
+ \frac{1}{3} \Lambda^2_{ijrs} \Lambda^2_{klqp}
+ \frac{1}{3} \Lambda^2_{ijsr} \Lambda^2_{klpq}
+ \frac{2}{3} \Lambda^2_{ijsr} \Lambda^2_{klqp} \\
& + & 2 \Lambda^2_{ikpr} \Lambda^2_{jlqs}
- \Lambda^2_{ikpq} \Lambda^2_{jlrs}
- \Lambda^2_{ikps} \Lambda^2_{jlqr}
- \Lambda^2_{ikqr} \Lambda^2_{jlps}
- \Lambda^2_{iksr} \Lambda^2_{jlqp} \\
& + & \frac{2}{3} \Lambda^2_{ikqs} \Lambda^2_{jlpr}
+ \frac{1}{3} \Lambda^2_{iksq} \Lambda^2_{jlpr}
+ \frac{1}{3} \Lambda^2_{ikqs} \Lambda^2_{jlrp}
+ \frac{2}{3} \Lambda^2_{iksq} \Lambda^2_{jlrp} \\
& + & 2 \Lambda^2_{ilps} \Lambda^2_{kjrq}
- \Lambda^2_{ilpr} \Lambda^2_{kjsq}
- \Lambda^2_{ilpq} \Lambda^2_{kjrs}
- \Lambda^2_{ilrs} \Lambda^2_{kjpq}
- \Lambda^2_{ilqs} \Lambda^2_{kjrp} \\
& + & \frac{2}{3} \Lambda^2_{ilrq} \Lambda^2_{kjps}
+ \frac{1}{3} \Lambda^2_{ilqr} \Lambda^2_{kjps}
+ \frac{1}{3} \Lambda^2_{ilrq} \Lambda^2_{kjsp}
+ \frac{2}{3} \Lambda^2_{ilqr} \Lambda^2_{kjsp}
\f}
By neglecting \f$ \Lambda^4 \f$, the cumulant approximation of the 4-RDM \f$ \Gamma^4 \f$ is obtained. \n
\n
[CUM1] M. Saitow, Y. Kurashige and T. Yanai, Journal of Chemical Physics 139, 044118 (2013). http://dx.doi.org/10.1063/1.4816627 \n*/
class Cumulant{
public:
//! Get the cumulant approximation of \f$ \Gamma^4_{ijklpqrs} \f$, using HAM indices
/** \param prob Pointer to the DMRG problem
\param the3DM Pointer to the DMRG 3-RDM
\param the2DM Pointer to the DMRG 2-RDM
\param i index 1 of \f$ \Gamma^4_{ijklpqrs} \f$
\param j index 2 of \f$ \Gamma^4_{ijklpqrs} \f$
\param k index 3 of \f$ \Gamma^4_{ijklpqrs} \f$
\param l index 4 of \f$ \Gamma^4_{ijklpqrs} \f$
\param p index 5 of \f$ \Gamma^4_{ijklpqrs} \f$
\param q index 6 of \f$ \Gamma^4_{ijklpqrs} \f$
\param r index 7 of \f$ \Gamma^4_{ijklpqrs} \f$
\param s index 8 of \f$ \Gamma^4_{ijklpqrs} \f$
\return the desired value */
static double gamma4_ham(const Problem * prob, const ThreeDM * the3DM, const TwoDM * the2DM, const int i, const int j, const int k, const int l, const int p, const int q, const int r, const int s);
//! Contract the CASPT2 Fock operator with the cumulant approximation of \f$ \Gamma^4 \f$ in \f$ \mathcal{O}(L^7) \f$ time, using HAM indices
/** \param prob Pointer to the DMRG problem
\param the3DM Pointer to the DMRG 3-RDM
\param the2DM Pointer to the DMRG 2-RDM
\param fock Contains the SYMMETRIC fock operator \f$ F_{ls} \f$ = fock[l+L*s] = fock[s+L*l]
\param result Contains the contraction: result[i+L*(j+L*(k+L*(p+L*(q+L*r))))] = \f$ \sum\limits_{ls} F_{ls} \Gamma^4_{ijklpqrs} \f$ */
static void gamma4_fock_contract_ham(const Problem * prob, const ThreeDM * the3DM, const TwoDM * the2DM, double * fock, double * result);
private:
// Get the second order cumulant \f$ \Lambda^2_{ijpq} \f$, using HAM indices
static double lambda2_ham(const TwoDM * the2DM, const int i, const int j, const int p, const int q);
};
}
#endif
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