CheMPS2
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CheMPS2::Correlations Class Reference

#include <Correlations.h>

Public Member Functions

 Correlations (const SyBookkeeper *denBKIn, const Problem *ProbIn, TwoDM *the2DMin)
 Constructor. More...
 
virtual ~Correlations ()
 Destructor.
 
double getCspin_DMRG (const int row, const int col) const
 Get a Cspin term, using the DMRG indices. More...
 
double getCspin_HAM (const int row, const int col) const
 Get a Cspin term, using the HAM indices. More...
 
double getCdens_DMRG (const int row, const int col) const
 Get a Cdens term, using the DMRG indices. More...
 
double getCdens_HAM (const int row, const int col) const
 Get a Cdens term, using the HAM indices. More...
 
double getCspinflip_DMRG (const int row, const int col) const
 Get a Cspinflip term, using the DMRG indices. More...
 
double getCspinflip_HAM (const int row, const int col) const
 Get a Cspinflip term, using the HAM indices. More...
 
double getCdirad_DMRG (const int row, const int col) const
 Get a Cdirad term, using the DMRG indices. More...
 
double getCdirad_HAM (const int row, const int col) const
 Get a Cdirad term, using the HAM indices. More...
 
double getMutualInformation_DMRG (const int row, const int col) const
 Get a mutual information term, using the DMRG indices. More...
 
double getMutualInformation_HAM (const int row, const int col) const
 Get a mutual information term, using the HAM indices. More...
 
double SingleOrbitalEntropy_DMRG (const int index) const
 Get the single-orbital entropy for a certain site, using the DMRG indices. More...
 
double SingleOrbitalEntropy_HAM (const int index) const
 Get the single-orbital entropy for a certain site, using hte HAM indices. More...
 
void FillSite (TensorT *denT, TensorGYZ **Gtensors, TensorGYZ **Ytensors, TensorGYZ **Ztensors, TensorKM **Ktensors, TensorKM **Mtensors)
 Fill at the current step of the iterations the two-orbital mutual information and the remaining part of Cdirad. More...
 
double MutualInformationDistance (const double power) const
 Return Idistance(power) (see return for definition) More...
 
void Print (const int precision=6, const int columnsPerLine=8) const
 Print the correlation functions and two-orbital mutual information. More...
 
void mpi_broadcast ()
 Broadcast the diradical correlation function and the two-orbital mutual information.
 

Detailed Description

Correlations class.

Author
Sebastian Wouters sebas.nosp@m.tian.nosp@m.woute.nosp@m.rs@g.nosp@m.mail..nosp@m.com
Date
August 9, 2014

Introduction

Orbital correlation can be quantified by means of the two-orbital mutual information, as well as several correlation functions. The two-orbital mutual information was introduced in quantum chemistry by Rissler [COR1] in order to find the optimal orbital ordering for DMRG calculations. Later, the groups of Legeza and Reiher used this quantum information measure to study both the orbital ordering, as well as the molecular electronic structure [COR2, COR3, COR4, COR5]. The two-orbital mutual information can be interpreted in terms of generalized correlation functions, which reveal the detailed nature of the electron correlation [COR6].

Two-orbital mutual information

In quantum chemistry, molecular electronic structure is often interpreted in terms of single-particle states, called orbitals. For correlated calculations, the orbital occupations become entangled, which is in contrast to Hartree-Fock theory, where the orbitals have definite occupations. Several measures from quantum information theory allow to quantify orbital correlation. From the total Hilbert space, a subset $A$ of orbitals can be selected. Denote by $B$ the complement of $A$, i.e. the other orbitals which form together with $A$ the total Hilbert space. From a wavefunction in the total Hilbert space $\ket{\Psi}$, the (positive semidefinite) reduced density matrix of $A$ can be calculated as

\[ \mathbf{\rho}_A = trace_{B} ~ \ket{\Psi}\bra{\Psi}, \]

where $trace_{B}$ denotes the summation over the occupations of the orbitals in $B$. The (nonnegative) eigenspectrum of $\mathbf{\rho}_A$ directly reflects the quantum entanglement between the orbitals in $A$ and the ones in $B$. If there is only one nonzero eigenvalue, $A$ and $B$ are not entangled. The total wavefunction can then be factorized: $\ket{\Psi} = \ket{\Psi_A}\ket{\Psi_B}$. A measurement of an occupation in $B$ does not influence the outcome of a measurement in $A$. If there are several nonzero eigenvalues, $A$ and $B$ are entangled, and measurements in $A$ and $B$ are correlated. The total wavefunction can not be factorized in that case.

The von Neumann entropy is a measure which quantifies the amount of entanglement between $A$ and $B$:

\[ S_{A \mid B} = -trace_A ~ \mathbf{\rho}_A \log( \mathbf{\rho}_A ) = S_{B \mid A} \geq 0. \]

If $A$ and $B$ are unentangled, $S_{A \mid B} = 0$. With increasing entanglement between $A$ and $B$, $S_{A \mid B}$ becomes larger. When $A$ consists of a single orbital $i$, the von Neumann entropy is called the single-orbital entropy $S_1(i)$. When $A$ consists of two orbitals $i$ and $j$, the von Neumann entropy is called the two-orbital entropy $S_2(ij)$. The following inequality holds between them:

\[ S_2(ij) \leq S_1(i) + S_1(j), \]

the so-called subadditivity property. Any entanglement between orbitals $i$ and $j$ reduces $S_2(ij)$ with respect to $S_1(i) + S_1(j)$. The amount of entanglement between two orbitals can then be quantified by means of the two-orbital mutual information

\[ \left[ \mathbf{I} \right]_{ij} = \frac{1}{2} \left( S_1(i) + S_1(j) - S_2(ij) \right) \left( 1 - \delta_{ij} \right) = \left[ \mathbf{I} \right]_{ji} \geq 0. \]

With increasing entanglement between orbitals $i$ and $j$, $\left[ \mathbf{I} \right]_{ij}$ becomes larger. In DMRG, this measure can be calculated efficiently [COR1, COR5, COR6]. In a spin-adapted DMRG code, there are only 16 unique elements instead of the 26 which are described in Ref. [COR5].

Correlation functions

In addition to the two-orbital mutual information, CheMPS2 also provides access to four other correlation functions. Three of them can be obtained from the reduced 2-RDMs $\Gamma^A$ and $\Gamma^B$, and from the reduced 1-RDM $\Gamma^1$:

\begin{eqnarray*} \Gamma^A_{ij;kl} & = & \sum\limits_{\sigma \tau} \braket{ \hat{a}_{i\sigma}^{\dagger} \hat{a}_{j\tau}^{\dagger} \hat{a}_{l\tau} \hat{a}_{k\sigma} }, \\ \Gamma^B_{ij;kl} & = & \sum\limits_{\sigma \tau} (-1)^{\sigma-\tau} \braket{ \hat{a}_{i\sigma}^{\dagger} \hat{a}_{j\tau}^{\dagger} \hat{a}_{l\tau} \hat{a}_{k\sigma} }, \\ \Gamma^1_{i;j} & = & \sum\limits_{\sigma} \braket{ \hat{a}_{i\sigma}^{\dagger} \hat{a}_{j\sigma} }, \end{eqnarray*}

where $\sigma$ and $\tau$ can be $\pm\frac{1}{2}$. These correlation functions are the spin correlation function

\[ C_{spin}(i,j) = 4 \left( \braket{\hat{S}_i^z \hat{S}_j^z} - \braket{\hat{S}_i^z} \braket{\hat{S}_j^z} \right) = \Gamma^B_{ij;ij} + \delta_{ij} \Gamma^1_{i;i}, \]

(where $\braket{\hat{S}_i^z}=0$ due to the spin-adaptation), the spin-flip correlation function

\[ C_{spinflip}(i,j) = \braket{\hat{S}_i^+ \hat{S}_j^-} + \braket{\hat{S}_i^- \hat{S}_j^+} = \frac{1}{2} \left( \Gamma^B_{ij;ji} - \Gamma^A_{ij;ji} \right) + \delta_{ij} \Gamma^1_{i;i}, \]

and the density correlation function

\[ C_{dens}(i,j) = \braket{\hat{n}_i \hat{n}_j} - \braket{\hat{n}_i} \braket{\hat{n}_j} = \Gamma^A_{ij;ij} + \Gamma^1_{i;i} \left( \delta_{ij} - \Gamma^1_{j;j} \right). \]

The fourth one is the singlet diradical correlation function [COR7]

\[ C_{dirad}(i,j) = \braket{\hat{d}_{i\uparrow} \hat{d}_{j\downarrow}} + \braket{\hat{d}_{i\downarrow} \hat{d}_{j\uparrow}} - \braket{\hat{d}_{i\uparrow}} \braket{\hat{d}_{j\downarrow}} - \braket{\hat{d}_{i\downarrow}} \braket{\hat{d}_{j\uparrow}}, \]

where $\hat{d}_{i\sigma} = \hat{n}_{i\sigma} (1 - \hat{n}_{i~-\sigma})$.

The two-orbital mutual information and these correlation functions are calculated in the function CheMPS2::DMRG::calc2DMandCorrelations.

References

[COR1] J. Rissler, R. M. Noack and S.R. White, Chemical Physics 323, 519-531 (2006). http://dx.doi.org/10.1016/j.chemphys.2005.10.018
[COR2] G. Barcza, O. Legeza, K.H. Marti, M. Reiher, Physical Review A 83, 012508 (2011). http://dx.doi.org/10.1103/PhysRevA.83.012508
[COR3] K. Boguslawski, K.H. Marti, O. Legeza and M. Reiher, Journal of Chemical Theory and Computation 8, 1970-1982 (2012). http://dx.doi.org/10.1021/ct300211j
[COR4] K. Boguslawski, P. Tecmer, O. Legeza and M. Reiher, Journal of Physical Chemistry Letters 3, 3129-3135 (2012). http://dx.doi.org/10.1021/jz301319v
[COR5] K. Boguslawski, P. Tecmer, G. Barcza, O. Legeza and M. Reiher, Journal of Chemical Theory and Computation 9, 2959-2973 (2013). http://dx.doi.org/10.1021/ct400247p
[COR6] G. Barcza, R.M. Noack, J. Solyom, O. Legeza, http://arxiv.org/abs/1406.6643 (2014).
[COR7] J. Hachmann, J.J. Dorando, M. Aviles and G.K.-L. Chan, Journal of Chemical Physics 127, 134309 (2007). http://dx.doi.org/10.1063/1.2768362

Definition at line 98 of file Correlations.h.

Constructor & Destructor Documentation

CheMPS2::Correlations::Correlations ( const SyBookkeeper *  denBKIn,
const Problem *  ProbIn,
TwoDM *  the2DMin 
)

Constructor.

Parameters
denBKInSymmetry sector bookkeeper
ProbInThe problem to be solved
the2DMinThe 2-RDM of the active space

Definition at line 35 of file Correlations.cpp.

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Member Function Documentation

void CheMPS2::Correlations::FillSite ( TensorT *  denT,
TensorGYZ **  Gtensors,
TensorGYZ **  Ytensors,
TensorGYZ **  Ztensors,
TensorKM **  Ktensors,
TensorKM **  Mtensors 
)

Fill at the current step of the iterations the two-orbital mutual information and the remaining part of Cdirad.

Parameters
denTDMRG site-matrices
GtensorsTensors required for the calculation
YtensorsTensors required for the calculation
ZtensorsTensors required for the calculation
KtensorsTensors required for the calculation
MtensorsTensors required for the calculation

Definition at line 212 of file Correlations.cpp.

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double CheMPS2::Correlations::getCdens_DMRG ( const int  row,
const int  col 
) const

Get a Cdens term, using the DMRG indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 117 of file Correlations.cpp.

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double CheMPS2::Correlations::getCdens_HAM ( const int  row,
const int  col 
) const

Get a Cdens term, using the HAM indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 119 of file Correlations.cpp.

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double CheMPS2::Correlations::getCdirad_DMRG ( const int  row,
const int  col 
) const

Get a Cdirad term, using the DMRG indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 141 of file Correlations.cpp.

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double CheMPS2::Correlations::getCdirad_HAM ( const int  row,
const int  col 
) const

Get a Cdirad term, using the HAM indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 143 of file Correlations.cpp.

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double CheMPS2::Correlations::getCspin_DMRG ( const int  row,
const int  col 
) const

Get a Cspin term, using the DMRG indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 105 of file Correlations.cpp.

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double CheMPS2::Correlations::getCspin_HAM ( const int  row,
const int  col 
) const

Get a Cspin term, using the HAM indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 107 of file Correlations.cpp.

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double CheMPS2::Correlations::getCspinflip_DMRG ( const int  row,
const int  col 
) const

Get a Cspinflip term, using the DMRG indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 129 of file Correlations.cpp.

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double CheMPS2::Correlations::getCspinflip_HAM ( const int  row,
const int  col 
) const

Get a Cspinflip term, using the HAM indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 131 of file Correlations.cpp.

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double CheMPS2::Correlations::getMutualInformation_DMRG ( const int  row,
const int  col 
) const

Get a mutual information term, using the DMRG indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 153 of file Correlations.cpp.

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double CheMPS2::Correlations::getMutualInformation_HAM ( const int  row,
const int  col 
) const

Get a mutual information term, using the HAM indices.

Parameters
rowthe first index
colthe second index
Returns
the desired value

Definition at line 155 of file Correlations.cpp.

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double CheMPS2::Correlations::MutualInformationDistance ( const double  power) const

Return Idistance(power) (see return for definition)

Parameters
powerThe power used in Idistance
Returns
$ Idistance(power) = sum_{ij} I(i,j) * \mid i-j \mid^{power} $

Definition at line 188 of file Correlations.cpp.

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void CheMPS2::Correlations::Print ( const int  precision = 6,
const int  columnsPerLine = 8 
) const

Print the correlation functions and two-orbital mutual information.

Parameters
precisionThe number of digits to be printed
columnsPerLineRarara: The number of columns per line

Definition at line 602 of file Correlations.cpp.

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double CheMPS2::Correlations::SingleOrbitalEntropy_DMRG ( const int  index) const

Get the single-orbital entropy for a certain site, using the DMRG indices.

Parameters
indexThe DMRG index
Returns
The single-orbital entropy for this site

Definition at line 165 of file Correlations.cpp.

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double CheMPS2::Correlations::SingleOrbitalEntropy_HAM ( const int  index) const

Get the single-orbital entropy for a certain site, using hte HAM indices.

Parameters
indexThe HAM index
Returns
The single-orbital entropy for this site

Definition at line 179 of file Correlations.cpp.

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