#include <CASSCF.h>

Public Member Functions
	CASSCF (Hamiltonian ham_in, int docc, int socc, int nocc, int ndmrg, int nvirt, const string tmp_folder=CheMPS2::defaultTMPpath)
	Constructor. More...

virtual	~CASSCF ()
	Destructor.

int	get_num_irreps ()
	Get the number of irreps. More...

double	solve (const int Nelectrons, const int TwoS, const int Irrep, ConvergenceScheme OptScheme, const int rootNum, DMRGSCFoptions scf_options)
	Do the CASSCF cycles with the augmented Hessian Newton-Raphson method. More...

double	caspt2 (const int Nelectrons, const int TwoS, const int Irrep, ConvergenceScheme OptScheme, const int rootNum, DMRGSCFoptions scf_options, const double IPEA, const double IMAG, const bool PSEUDOCANONICAL, const bool CHECKPOINT=false, const bool CUMULANT=false)
	Calculate the caspt2 correction energy for a converged casscf wavefunction. More...

Static Public Member Functions
static void	deleteStoredUnitary (const string filename=CheMPS2::DMRGSCF_unitary_storage_name)
	CASSCF unitary rotation remove call.

static void	deleteStoredDIIS (const string filename=CheMPS2::DMRGSCF_diis_storage_name)
	CASSCF DIIS vectors remove call.

static void	buildFmat (DMRGSCFmatrix localFmat, const DMRGSCFmatrix localTmat, const DMRGSCFmatrix localJKocc, const DMRGSCFmatrix localJKact, const DMRGSCFindices localIdx, const DMRGSCFintegrals theInts, double local2DM, double local1DM)
	Build the F-matrix (Eq. (11) in the Siegbahn paper [CAS3]) More...

static void	buildWtilde (DMRGSCFwtilde localwtilde, const DMRGSCFmatrix localTmat, const DMRGSCFmatrix localJKocc, const DMRGSCFmatrix localJKact, const DMRGSCFindices localIdx, const DMRGSCFintegrals theInts, double local2DM, double local1DM)
	Build the Wtilde-matrix (Eq. (20b) in the Siegbahn paper [CAS3]) More...

static void	augmentedHessianNR (DMRGSCFmatrix localFmat, DMRGSCFwtilde localwtilde, const DMRGSCFindices localIdx, const DMRGSCFunitary localUmat, double theupdate, double updateNorm, double *gradNorm)
	Calculate the augmented Hessian Newton-Raphson update for the orthogonal orbital rotation matrix. More...

static void	copy2DMover (TwoDM theDMRG2DM, const int LAS, double two_dm)
	Copy over the DMRG 2-RDM. More...

static void	setDMRG1DM (const int num_elec, const int LAS, double one_dm, double two_dm)
	Construct the 1-RDM from the 2-RDM. More...

static void	copy_active (double origin, DMRGSCFmatrix result, const DMRGSCFindices *idx, const bool one_rdm)
	Copy a one-orbital quantity from array format to DMRGSCFmatrix format. More...

static void	copy_active (const DMRGSCFmatrix origin, double result, const DMRGSCFindices *idx)
	Copy a one-orbital quantity from DMRGSCFmatrix format to array format. More...

static void	fillLocalizedOrbitalRotations (DMRGSCFunitary umat, DMRGSCFindices idx, double *eigenvecs)
	From an Edmiston-Ruedenberg active space rotation, fetch the eigenvectors and store them in eigenvecs. More...

static void	block_diagonalize (const char space, const DMRGSCFmatrix Mat, DMRGSCFunitary Umat, double work1, double work2, const DMRGSCFindices idx, const bool invert, double two_dm, double three_dm, double contract)
	Block-diagonalize Mat. More...

static void	construct_fock (DMRGSCFmatrix Fock, const DMRGSCFmatrix Tmat, const DMRGSCFmatrix Qocc, const DMRGSCFmatrix Qact, const DMRGSCFindices *idx)
	Construct the Fock matrix. More...

static double	deviation_from_blockdiag (DMRGSCFmatrix matrix, const DMRGSCFindices idx)
	Return the RMS deviation from block-diagonal. More...

static void	write_f4rdm_checkpoint (const string f4rdm_file, int hamorb1, int hamorb2, const int tot_dmrg_power6, double *contract)
	Write the checkpoint file for the contraction of the generalized Fock operator with the 4-RDM to disk. More...

static bool	read_f4rdm_checkpoint (const string f4rdm_file, int hamorb1, int hamorb2, const int tot_dmrg_power6, double *contract)
	Read the checkpoint file for the contraction of the generalized Fock operator with the 4-RDM from disk. More...

static void	fock_dot_4rdm (double fockmx, CheMPS2::DMRG dmrgsolver, CheMPS2::Hamiltonian ham, int next_orb1, int next_orb2, double work, double *result, const bool CHECKPOINT, const bool PSEUDOCANONICAL)
	Build the contraction of the fock matrix with the 4-RDM. More...

Detailed Description

CASSCF class.

Author: Sebastian Wouters sebas.nosp@m.tian.nosp@m.woute.nosp@m.rs@g.nosp@m.mail..nosp@m.com

Date: June 18, 2013

Introduction

In methods which use a FCI solver, this solver can be replaced by DMRG. DMRG allows for an efficient extraction of the 2-RDM [CAS1, CAS2]. The 2-RDM of the active space is required in the complete active space self-consistent field (CASSCF) method to compute the gradient and the Hessian with respect to orbital rotations [CAS3]. It is therefore natural to introduce a CASSCF variant with DMRG as active space solver, called DMRG-SCF [CAS2, CAS4, CAS5], which allows to treat static correlation in large active spaces. In CheMPS2, I have implemented the augmented Hessian Newton-Raphson DMRG-SCF method, with exact Hessian [CAS5, CAS6]. It can be called with the function CheMPS2::CASSCF::doCASSCFnewtonraphson.

Orbital gradient and Hessian

The calculation of the orbital gradient and Hessian for DMRG-SCF is based on [CAS3]. The basic idea is to express the energy with the unitary group generators:

$\begin{eqnarray*} \hat{E}_{pq} & = & \sum\limits_{\sigma} \hat{a}^{\dagger}_{p \sigma} \hat{a}_{q \sigma} \\ \left[ \hat{E}_{pq} , \hat{E}_{rs} \right] & = & \delta_{qr} \hat{E}_{ps} - \delta_{ps} \hat{E}_{rq} \\ \hat{E}^{-}_{pq} & = & \hat{E}_{pq} - \hat{E}_{qp} \\ \hat{T} & = & \sum\limits_{p<q} x_{pq} \hat{E}^{-}_{pq} \\ E(\vec{x}) & = & \braket{0 | e^{\hat{T}} \hat{H} e^{-\hat{T}} | 0 } \\ \left. \frac{\partial E(\vec{x})}{\partial x_{ij}} \right|_{0} & = & \braket{ 0 | \left[ \hat{E}_{ij}^{-}, \hat{H} \right] | 0 } \\ \left. \frac{\partial^2 E(\vec{x})}{\partial x_{ij} \partial x_{kl}} \right|_{0} & = & \frac{1}{2} \braket{ 0 | \left[ \hat{E}_{ij}^{-}, \left[ \hat{E}_{kl}^{-}, \hat{H} \right] \right] | 0 } + \frac{1}{2} \braket{ 0 | \left[ \hat{E}_{kl}^{-}, \left[ \hat{E}_{ij}^{-}, \hat{H} \right] \right] | 0 } \end{eqnarray*}$

The variables $x_{pq}$ only connect orbitals with the same irrep ( $I_p=I_q$ ). Assuming that DMRG is exact, $x_{pq}$ in addition only connects orbitals when they belong to different occupation blocks: occupied, active, virtual. With some algebra, the derivatives can be rewritten. Real-valued symmetric one-electron integrals $h_{ij}$ and real-valued eightfold permutation symmetric two-electron integrals $(ij | kl)$ are assumed (chemical notation for the latter).

$\begin{eqnarray*} \Gamma^{2A}_{ijkl} & = & \sum\limits_{\sigma \tau} \braket{ 0 | \hat{a}^{\dagger}_{i \sigma} \hat{a}_{j \tau}^{\dagger} \hat{a}_{l \tau} \hat{a}_{k \sigma} | 0} \\ \Gamma^1_{ij} & = & \sum\limits_{\sigma} \braket{ 0 | \hat{a}^{\dagger}_{i \sigma} \hat{a}_{j \sigma} | 0} \\ \left. \frac{\partial E(\vec{x})}{\partial x_{ij}} \right|_{0} & = & 2 \left( F_{ij} - F_{ji} \right) \\ F_{pq} & = & \sum\limits_{r} \Gamma_{pr}^{1} h_{qr} + \sum\limits_{rst} \Gamma^{2A}_{psrt} (qr | st) \\ \left. \frac{\partial^2 E(\vec{x})}{\partial x_{ij} \partial x_{kl}} \right|_{0} & = & w_{ijkl} - w_{jikl} - w_{ijlk} + w_{jilk} \\ w_{pqrs} & = & \delta_{qr} \left( F_{ps} + F_{sp} \right) + \tilde{w}_{pqrs} \\ \tilde{w}_{pqrs} & = & 2 \Gamma^1_{pr} h_{qs} + 2 \sum\limits_{\alpha \beta} \left( \Gamma^{2A}_{r \alpha p \beta} (qs | \alpha \beta ) + \left( \Gamma^{2A}_{r \alpha \beta p} + \Gamma^{2A}_{r p \beta \alpha} \right) (q \alpha | s \beta ) \right) \end{eqnarray*}$

In the calculation of $F_{pq}$ , the indices $prst$ can only be occupied or active due to their appearance in the density matrices, and the only index which can be virtual is hence $q$ . Moreover, due to the irrep symmetry of the integrals and density matrices, $F_{pq}$ is diagonal in the irreps: $I_p = I_q$ . Alternatively, this can be understood by the fact that $x_{pq}$ only connects orbitals with the same irrep.

In the calculation of $\tilde{w}_{pqrs}$ , the indices $pr\alpha\beta$ can only be occupied or active due to their appearance in the density matrices, and the only indices which can be virtual are hence $qs$ . Together with the remark for $F_{pq}$ , this can save time for the two-electron integral rotation. Moreover, as $x_{pq}$ only connects orbitals with the same irrep, $I_p = I_q$ and $I_r = I_s$ in $\tilde{w}_{pqrs}$ .

By rewriting the density matrices, the calculation of $F_{pq}$ and $\tilde{w}_{pqrs}$ can be simplified. In the following, occ and act denote the doubly occupied and active orbital spaces, respectively.

$\begin{eqnarray*} \Gamma^1_{ij} & = & 2 \delta_{ij}^{occ} + \Gamma^{1,act}_{ij} \\ \Gamma^{2A}_{ijkl} & = & \Gamma^{2A,act}_{ijkl} + 2 \delta_{ik}^{occ} \Gamma^{1,act}_{jl} - \delta_{il}^{occ} \Gamma^{1,act}_{jk} \\ & + & 2 \delta_{jl}^{occ} \Gamma^{1,act}_{ik} - \delta_{jk}^{occ} \Gamma^{1,act}_{il} + 4 \delta_{ik}^{occ} \delta_{jl}^{occ} - 2 \delta_{il}^{occ} \delta_{jk}^{occ} \end{eqnarray*}$

Define the following symmetric charge (Coulomb + exchange) matrices:

$\begin{eqnarray*} Q^{occ}_{ij} & = & \sum\limits_{s \in occ} \left[ 2 (ij | ss) - (is | js ) \right] \\ Q^{act}_{ij} & = & \sum\limits_{st \in act} \frac{1}{2} \Gamma^{1,act}_{st} \left[ 2 (ij | st) - (is | jt ) \right] \end{eqnarray*}$

They can be calculated efficiently by (1) rotating the occupied and active density matrices from the current basis to the original basis, (2) contracting the rotated density matrices with the two-electron integrals in the original basis, and (3) rotating these contractions to the current basis. The constant part and the one-electron integrals of the active space Hamiltonian are:

$\begin{eqnarray*} \tilde{E}_{const} & = & E_{const} + \sum\limits_{s \in occ} \left( 2 h_{ss} + Q_{ss}^{occ} \right) \\ \tilde{h}_{ij} & = & h_{ij} + Q_{ij}^{occ} \end{eqnarray*}$

The calculation of $F_{pq}$ boils down to:

$\begin{eqnarray*} p \in occ & : & F_{pq} = 2 \left( h_{qp} + Q^{occ}_{qp} + Q^{act}_{qp} \right) \\ p \in act & : & F_{pq} = \sum\limits_{r \in act} \Gamma^{1,act}_{pr} \left[ h_{qr} + Q^{occ}_{qr} \right] + \sum\limits_{rst \in act} \Gamma^{2A,act}_{psrt} (qr | st) \end{eqnarray*}$

And the calculation of $\tilde{w}_{pqrs}$ (remember that $I_p = I_q$ and $I_r = I_s$ ):

$\begin{eqnarray*} (p,r) & \in & (occ,occ) : \tilde{w}_{pqrs} \\ & = & 4 \delta_{pr}^{occ} \left[ h_{qs} + Q^{occ}_{qs} + Q^{act}_{qs} \right] \\ & + & 4 \left[ 4 (qp | sr) - ( qs | pr ) - ( qr | sp ) \right] \\ (p,r) & \in & (act,act) : \tilde{w}_{pqrs} = 2 \Gamma^{1,act}_{rp} \left[ h_{qs} + Q^{occ}_{qs} \right] \\ & + & 2 \sum\limits_{\alpha\beta \in act} \left[ \Gamma^{2A,act}_{r \alpha p \beta} (qs | \alpha \beta ) + \left( \Gamma^{2A,act}_{r \alpha \beta p} + \Gamma^{2A,act}_{r p \beta \alpha} \right) (q \alpha | s \beta ) \right] \\ (p,r) & \in & (act,occ) : \tilde{w}_{pqrs} \\ & = & 2 \sum\limits_{\alpha \in act} \Gamma^{1,act}_{\alpha p} \left[ 4 (q \alpha | s r) - (qs | \alpha r) - (qr | s \alpha) \right] \\ (p,r) & \in & (occ,act) : \tilde{w}_{pqrs} \\ & = & 2 \sum\limits_{\beta \in act} \Gamma^{1,act}_{r \beta} \left[ 4 (q p | s \beta) - (qs | p \beta) - (q \beta | sp) \right] \end{eqnarray*}$

Augmented Hessian Newton-Raphson DMRG-SCF

The CASSCF energy is a function of $\vec{x}$ . Up to second order, the energy is given by

$E(\vec{x}) = \braket{0 | e^{\hat{T}(\vec{x})} \hat{H} e^{-\hat{T}(\vec{x})} | 0 } \approx E(0) + \vec{x}^T \vec{g} + \frac{1}{2} \vec{x}^T \mathbf{H} \vec{x}.$

The vector $\vec{g}$ is the gradient and the matrix $\mathbf{H}$ the Hessian for orbital rotations [CAS3]. They have been described in the previous section. The minimum of $E(\vec{x})$ is found at $\vec{x} = - \mathbf{H}^{-1} \vec{g}$ . The variables $\vec{x}$ parametrize an additional orbital rotation $\mathbf{U}_{add} = \exp(\mathbf{X}(\vec{x}))$ , with $\mathbf{X}(\vec{x}) = -\mathbf{X}^T(\vec{x})$ a real-valued skew-symmetric matrix. The additional orbital rotation $\mathbf{U}_{add}$ transforms the current orbitals $\mathbf{U}(n)$ to the new orbitals

$\mathbf{U}(n+1) = \mathbf{U}_{add} \mathbf{U}(n) = \exp(\mathbf{X}(\vec{x}(n))) \mathbf{U}(n).$

This updating scheme is called the Newton-Raphson method [CAS3]. If the Hessian is positive definite, these updates are stable. For saddle points in the energy landscape, the Hessian has negative eigenvalues, and these updates can be unstable. It is therefore better to use the augmented Hessian Newton-Raphson method [CAS7]:

$\left[ \begin{array}{cc} \mathbf{H} & \vec{g} \\ \vec{g}^T & 0 \end{array} \right] \left[ \begin{array}{c} \vec{x} \\ 1 \end{array} \right] = \alpha \left[ \begin{array}{c} \vec{x} \\ 1 \end{array} \right].$

The eigenvector with smallest algebraic eigenvalue determines a stable update $\vec{x}$ , as is well explained in Ref. [CAS7].

As a final remark in this section, I would like to say that orbitals have gauge freedom. One can always multiply them with a phase factor. It is therefore possible to choose the orbital gauges so that all $\mathbf{U}$ are always special orthogonal: $\det(\mathbf{U})=+1$ .

Direct inversion of the iterative subspace (DIIS)

When the update norm $\|\vec{x}\|_2$ is small enough, the convergence can be accelerated by the direct inversion of the iterative subspace (DIIS) [CAS5, CAS8, CAS9, CAS10]. For a given set of orbitals $\mathbf{U}(n)$ , the update $\vec{x}(n)$ is calculated with the augmented Hessian Newton-Raphson method. This update defines the next set of orbitals:

$\mathbf{U}(n+1) = \mathbf{U}_{add} \mathbf{U}(n) = \exp(\mathbf{X}(\vec{x}(n))) \mathbf{U}(n).$

In DIIS, the error vector $\vec{x}(n)$ and the state vector $\mathbf{Y}(n+1) = \log(\mathbf{U}(n+1))$ are added to a list. The error

$e = \left\| \sum\limits_{i = 1}^{\kappa} c_i \vec{x}(n - \kappa + i) \right\|_2$

is minimized under the constraint $\sum_{i} c_i = 1$ . $\kappa$ is the size of the list memory, i.e. the number of retained vectors. The minimization of the error $e$ can be performed with Lagrangian calculus:

$\left[ \begin{array}{cc} \mathbf{B} & \vec{1} \\ \vec{1}^T & 0 \end{array} \right] \left[ \begin{array}{c} \vec{c} \\ \lambda \end{array} \right] = \left[ \begin{array}{c} \vec{0} \\ 1 \end{array} \right],$

where $2\lambda$ is the Lagrangian multiplier and

$\left[\mathbf{B}\right]_{ij} = \vec{x}^T(n - \kappa + i) \vec{x}(n - \kappa + j) = \left[\mathbf{B}\right]_{ji}.$

The new state vector is then defined as

$\mathbf{Y}_{new} = \sum\limits_{i = 1}^{\kappa} c_i \mathbf{Y}(n+1 - \kappa + i).$

The new state vector $\mathbf{Y}_{new}$ is calculated by the function CheMPS2::DIIS::calculateParam. The current orbitals are then set to $\mathbf{U}(n+1) = \exp(\mathbf{Y}_{new})$ .

References

[CAS1] D. Zgid and M. Nooijen, Journal of Chemical Physics 128, 144115 (2008). http://dx.doi.org/10.1063/1.2883980
[CAS2] D. Ghosh, J. Hachmann, T. Yanai and G.K.-L. Chan, Journal of Chemical Physics 128, 144117 (2008). http://dx.doi.org/10.1063/1.2883976
[CAS3] P.E.M. Siegbahn, J. Almlof, A. Heiberg and B.O. Roos, Journal of Chemical Physics 74, 2384-2396 (1981). http://dx.doi.org/10.1063/1.441359
[CAS4] D. Zgid and M. Nooijen, Journal of Chemical Physics 128, 144116 (2008). http://dx.doi.org/10.1063/1.2883981
[CAS5] T. Yanai, Y. Kurashige, D. Ghosh and G.K.-L. Chan, International Journal of Quantum Chemistry 109, 2178-2190 (2009). http://dx.doi.org/10.1002/qua.22099
[CAS6] S. Wouters, W. Poelmans, P.W. Ayers and D. Van Neck, Computer Physics Communications 185, 1501-1514 (2014). http://dx.doi.org/10.1016/j.cpc.2014.01.019
[CAS7] A. Banerjee, N. Adams, J. Simons and R. Shepard, Journal of Physical Chemistry 89, 52-57 (1985). http://dx.doi.org/10.1021/j100247a015
[CAS8] P. Pulay, Chemical Physics Letters 73, 393-398 (1980). http://dx.doi.org/10.1016/0009-2614(80)80396-4
[CAS9] C.D. Sherrill, Programming DIIS, http://vergil.chemistry.gatech.edu/notes/diis/node3.html (2000).
[CAS10] T. Rohwedder and R. Schneider, Journal of Mathematical Chemistry 49, 1889-1914 (2011). http://dx.doi.org/10.1007/s10910-011-9863-y

Definition at line 164 of file CASSCF.h.

Constructor & Destructor Documentation

CheMPS2::CASSCF::CASSCF	(	Hamiltonian *	ham_in,
		int *	docc,
		int *	socc,
		int *	nocc,
		int *	ndmrg,
		int *	nvirt,
		const string	tmp_folder = `CheMPS2::defaultTMPpath`
	)

Constructor.

Parameters

ham_in	Hamiltonian containing the matrix elements of the Hamiltonian for which a CASSCF calculation is desired
docc	Array containing the number of doubly occupied HF orbitals per irrep
socc	Array containing the number of singly occupied HF orbitals per irrep
nocc	Array containing the number of doubly occupied (inactive) orbitals per irrep
ndmrg	Array containing the number of active orbitals per irrep
nvirt	Array containing the number of virtual (secondary) orbitals per irrep
tmp_folder	Temporary work folder for the DMRG renormalized operators and the ERI rotations

Definition at line 39 of file CASSCF.cpp.

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Member Function Documentation

void CheMPS2::CASSCF::augmentedHessianNR	(	DMRGSCFmatrix *	localFmat,
		DMRGSCFwtilde *	localwtilde,
		const DMRGSCFindices *	localIdx,
		const DMRGSCFunitary *	localUmat,
		double *	theupdate,
		double *	updateNorm,
		double *	gradNorm
	)

static

Calculate the augmented Hessian Newton-Raphson update for the orthogonal orbital rotation matrix.

Parameters

localFmat	Matrix which contains the Fock operator (Eq. (11) in the Siegbahn paper [CAS3])
localwtilde	Object which contains the second order derivative of the energy with respect to the unitary (Eq. (20b) in the Siegbahn paper [CAS3])
localIdx	Orbital index bookkeeper for the CASSCF calculations
localUmat	The unitary matrix for CASSCF calculations (in this function it is used to fetch the orbital ordering convention of the skew-symmetric parametrization)
theupdate	Where the augmented Hessian Newton-Raphson update will be stored
updateNorm	Pointer to one double to store the update norm
gradNorm	Pointer to one double to store the gradient norm

Definition at line 316 of file CASSCFnewtonraphson.cpp.

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void CheMPS2::CASSCF::block_diagonalize	(	const char	space,
		const DMRGSCFmatrix *	Mat,
		DMRGSCFunitary *	Umat,
		double *	work1,
		double *	work2,
		const DMRGSCFindices *	idx,
		const bool	invert,
		double *	two_dm,
		double *	three_dm,
		double *	contract
	)

static

Block-diagonalize Mat.

Parameters

space	Can be 'O', 'A', or 'V' and denotes which block of Mat should be considered
Mat	Matrix to block-diagonalize
Umat	The unitary rotation will be updated so that Mat is block-diagonal in the orbitals 'space'
work1	Workspace
work2	Workspace
idx	Object which handles the index conventions for CASSCF
invert	If true, the eigenvectors are sorted from large to small instead of the other way around
two_dm	If not NULL, this 4-index array will be rotated to the new eigenvecs if space == 'A'
three_dm	If not NULL, this 6-index array will be rotated to the new eigenvecs if space == 'A'
contract	If not NULL, this 6-index array will be rotated to the new eigenvecs if space == 'A'

Definition at line 433 of file CASSCF.cpp.

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void CheMPS2::CASSCF::buildFmat	(	DMRGSCFmatrix *	localFmat,
		const DMRGSCFmatrix *	localTmat,
		const DMRGSCFmatrix *	localJKocc,
		const DMRGSCFmatrix *	localJKact,
		const DMRGSCFindices *	localIdx,
		const DMRGSCFintegrals *	theInts,
		double *	local2DM,
		double *	local1DM
	)

static

Build the F-matrix (Eq. (11) in the Siegbahn paper [CAS3])

Parameters

localFmat	Matrix where the result should be stored
localTmat	Matrix which contains the one-electron integrals
localJKocc	Matrix which contains the Coulomb and exchange interaction due to the frozen core orbitals
localJKact	Matrix which contains the Coulomb and exchange interaction due to the active space
localIdx	Orbital index bookkeeper for the CASSCF calculations
theInts	The rotated two-electron integrals (at most 2 virtual indices)
local2DM	The DMRG 2-RDM
local1DM	The DMRG 1-RDM

Definition at line 823 of file CASSCFnewtonraphson.cpp.

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void CheMPS2::CASSCF::buildWtilde	(	DMRGSCFwtilde *	localwtilde,
		const DMRGSCFmatrix *	localTmat,
		const DMRGSCFmatrix *	localJKocc,
		const DMRGSCFmatrix *	localJKact,
		const DMRGSCFindices *	localIdx,
		const DMRGSCFintegrals *	theInts,
		double *	local2DM,
		double *	local1DM
	)

static

Build the Wtilde-matrix (Eq. (20b) in the Siegbahn paper [CAS3])

Parameters

localwtilde	Where the result should be stored
localTmat	Matrix which contains the one-electron integrals
localJKocc	Matrix which contains the Coulomb and exchange interaction due to the frozen core orbitals
localJKact	Matrix which contains the Coulomb and exchange interaction due to the active space
localIdx	Orbital index bookkeeper for the CASSCF calculations
theInts	The rotated two-electron integrals (at most 2 virtual indices)
local2DM	The DMRG 2-RDM
local1DM	The DMRG 1-RDM

Definition at line 611 of file CASSCFnewtonraphson.cpp.

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double CheMPS2::CASSCF::caspt2	(	const int	Nelectrons,
		const int	TwoS,
		const int	Irrep,
		ConvergenceScheme *	OptScheme,
		const int	rootNum,
		DMRGSCFoptions *	scf_options,
		const double	IPEA,
		const double	IMAG,
		const bool	PSEUDOCANONICAL,
		const bool	CHECKPOINT = `false`,
		const bool	CUMULANT = `false`
	)

Calculate the caspt2 correction energy for a converged casscf wavefunction.

Parameters

Nelectrons	Total number of electrons in the system: occupied HF orbitals + active space
TwoS	Twice the targeted spin
Irrep	Desired wave-function irrep
OptScheme	The optimization scheme to run the inner DMRG loop. If NULL: use FCI instead of DMRG.
rootNum	Denotes the targeted state in state-specific CASSCF; 1 means ground state, 2 first excited state etc.
scf_options	Contains the DMRGSCF options
IPEA	The CASPT2 IPEA shift from Ghigo, Roos and Malmqvist, Chemical Physics Letters 396, 142-149 (2004)
IMAG	The CASPT2 imaginary shift from Forsberg and Malmqvist, Chemical Physics Letters 274, 196-204 (1997)
PSEUDOCANONICAL	If true, use the exact DMRG 4-RDM in the pseudocanonical basis. If false, use the cumulant approximated DMRG 4-RDM in the unrotated basis.
CHECKPOINT	If true, write checkpoints to disk and read them back in again in order to perform the contraction of the generalized Fock operator with the 4-RDM in multiple runs.
CUMULANT	If true, a cumulant approximation is used for the 4-RDM and CHECKPOINT is overwritten to false. If false, the full 4-RDM is used.

Returns: The CASPT2 variational correction energy

Definition at line 189 of file CASSCFpt2.cpp.

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void CheMPS2::CASSCF::construct_fock	(	DMRGSCFmatrix *	Fock,
		const DMRGSCFmatrix *	Tmat,
		const DMRGSCFmatrix *	Qocc,
		const DMRGSCFmatrix *	Qact,
		const DMRGSCFindices *	idx
	)

static

Construct the Fock matrix.

Parameters

Fock	Matrix to store the Fock operator in
Tmat	Matrix with the one-electron integrals
Qocc	Matrix with the Coulomb and exchange contributions of the occupied (inactive) orbitals
Qact	Matrix with the Coulomb and exchange contributions of the active space orbitals
idx	Object which handles the index conventions for CASSCF

Definition at line 395 of file CASSCFpt2.cpp.

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void CheMPS2::CASSCF::copy2DMover	(	TwoDM *	theDMRG2DM,
		const int	LAS,
		double *	two_dm
	)

static

Copy over the DMRG 2-RDM.

Parameters

theDMRG2DM	The 2-RDM from the DMRG object
LAS	The total number of DMRG orbitals
two_dm	The CASSCF 2-RDM

Definition at line 119 of file CASSCF.cpp.

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void CheMPS2::CASSCF::copy_active	(	double *	origin,
		DMRGSCFmatrix *	result,
		const DMRGSCFindices *	idx,
		const bool	one_rdm
	)

static

Copy a one-orbital quantity from array format to DMRGSCFmatrix format.

Parameters

origin	Array to copy
result	DMRGSCFmatrix to store the copy
idx	Object which handles the index conventions for CASSCF
one_rdm	If true, the occupied orbitals get occupation 2

Definition at line 375 of file CASSCF.cpp.

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void CheMPS2::CASSCF::copy_active	(	const DMRGSCFmatrix *	origin,
		double *	result,
		const DMRGSCFindices *	idx
	)

static

Copy a one-orbital quantity from DMRGSCFmatrix format to array format.

Parameters

origin	DMRGSCFmatrix to copy
result	Array to store the copy
idx	Object which handles the index conventions for CASSCF

Definition at line 407 of file CASSCF.cpp.

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double CheMPS2::CASSCF::deviation_from_blockdiag	(	DMRGSCFmatrix *	matrix,
		const DMRGSCFindices *	idx
	)

static

Return the RMS deviation from block-diagonal.

Parameters

matrix	Matrix to be assessed
idx	Object which handles the index conventions for CASSCF

Returns: RMS deviation from block-diagonal

Definition at line 411 of file CASSCFpt2.cpp.

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void CheMPS2::CASSCF::fillLocalizedOrbitalRotations	(	DMRGSCFunitary *	umat,
		DMRGSCFindices *	idx,
		double *	eigenvecs
	)

static

From an Edmiston-Ruedenberg active space rotation, fetch the eigenvectors and store them in eigenvecs.

Parameters

umat	The Edmiston-Ruedenberg active space rotation
idx	Object which handles the index conventions for CASSCF
eigenvecs	Where the eigenvectors are stored

Definition at line 160 of file CASSCF.cpp.

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void CheMPS2::CASSCF::fock_dot_4rdm	(	double *	fockmx,
		CheMPS2::DMRG *	dmrgsolver,
		CheMPS2::Hamiltonian *	ham,
		int	next_orb1,
		int	next_orb2,
		double *	work,
		double *	result,
		const bool	CHECKPOINT,
		const bool	PSEUDOCANONICAL
	)

static

Build the contraction of the fock matrix with the 4-RDM.

Parameters

fockmx	Array of size ham->getL() x ham->getL() containing the Fock matrix elements
dmrgsolver	DMRG object which is solved, and for which the 2-RDM and 3-RDM have been calculated as well
ham	Active space Hamiltonian, which is needed for the size of the active space and the orbital irreps
next_orb1	The next first orbital for which the contraction should be continued
next_orb2	The next second orbital for which the contraction should be continued
work	Work array of size ham->getL()**6
result	On entry, contains the partial contraction corresponding to (next_orb1, next_orb2). On exit, contains the full contraction.
CHECKPOINT	Whether or not the standard CheMPS2 F.4-RDM checkpoint should be created/updated to continue the contraction at later times
PSEUDOCANONICAL	Whether or not pseudocanonical orbitals are used in the active space

Definition at line 140 of file CASSCFpt2.cpp.

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int CheMPS2::CASSCF::get_num_irreps ( )

Get the number of irreps.

Returns: The number of irreps

Definition at line 117 of file CASSCF.cpp.

bool CheMPS2::CASSCF::read_f4rdm_checkpoint	(	const string	f4rdm_file,
		int *	hamorb1,
		int *	hamorb2,
		const int	tot_dmrg_power6,
		double *	contract
	)

static

Read the checkpoint file for the contraction of the generalized Fock operator with the 4-RDM from disk.

Parameters

f4rdm_file	The filename
hamorb1	The next hamiltonian orbital 1
hamorb2	The next hamiltonian orbital 2
tot_dmrg_power6	The size of the array contract
contract	The current partial contraction

Returns: Whether the file was found and read

Definition at line 87 of file CASSCFpt2.cpp.

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void CheMPS2::CASSCF::setDMRG1DM	(	const int	num_elec,
		const int	LAS,
		double *	one_dm,
		double *	two_dm
	)

static

Construct the 1-RDM from the 2-RDM.

Parameters

num_elec	The number of DMRG active space electrons
LAS	The total number of DMRG orbitals
one_dm	The CASSCF 1-RDM
two_dm	The CASSCF 2-RDM

Definition at line 134 of file CASSCF.cpp.

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double CheMPS2::CASSCF::solve	(	const int	Nelectrons,
		const int	TwoS,
		const int	Irrep,
		ConvergenceScheme *	OptScheme,
		const int	rootNum,
		DMRGSCFoptions *	scf_options
	)

Do the CASSCF cycles with the augmented Hessian Newton-Raphson method.

Parameters

Nelectrons	Total number of electrons in the system: occupied HF orbitals + active space
TwoS	Twice the targeted spin
Irrep	Desired wave-function irrep
OptScheme	The optimization scheme to run the inner DMRG loop. If NULL: use FCI instead of DMRG.
rootNum	Denotes the targeted state in state-specific CASSCF; 1 means ground state, 2 first excited state etc.
scf_options	Contains the DMRGSCF options

Returns: The converged DMRGSCF energy

Definition at line 56 of file CASSCFnewtonraphson.cpp.

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void CheMPS2::CASSCF::write_f4rdm_checkpoint	(	const string	f4rdm_file,
		int *	hamorb1,
		int *	hamorb2,
		const int	tot_dmrg_power6,
		double *	contract
	)

static

Write the checkpoint file for the contraction of the generalized Fock operator with the 4-RDM to disk.

Parameters

f4rdm_file	The filename
hamorb1	The next hamiltonian orbital 1
hamorb2	The next hamiltonian orbital 2
tot_dmrg_power6	The size of the array contract
contract	The current partial contraction

Definition at line 44 of file CASSCFpt2.cpp.

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The documentation for this class was generated from the following files:

Public Member Functions

Static Public Member Functions

Detailed Description

Introduction

Orbital gradient and Hessian

Augmented Hessian Newton-Raphson DMRG-SCF

Direct inversion of the iterative subspace (DIIS)

References

Constructor & Destructor Documentation

Member Function Documentation