# A general non-Abelian density matrix renormalization group algorithm with application to the C{sub 2} dimer

## Abstract

We extend our previous work [S. Sharma and G. K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)], which described a spin-adapted (SU(2) symmetry) density matrix renormalization group algorithm, to additionally utilize general non-Abelian point group symmetries. A key strength of the present formulation is that the requisite tensor operators are not hard-coded for each symmetry group, but are instead generated on the fly using the appropriate Clebsch-Gordan coefficients. This allows our single implementation to easily enable (or disable) any non-Abelian point group symmetry (including SU(2) spin symmetry). We use our implementation to compute the ground state potential energy curve of the C{sub 2} dimer in the cc-pVQZ basis set (with a frozen-core), corresponding to a Hilbert space dimension of 10{sup 12} many-body states. While our calculated energy lies within the 0.3 mE{sub h} error bound of previous initiator full configuration interaction quantum Monte Carlo and correlation energy extrapolation by intrinsic scaling calculations, our estimated residual error is only 0.01 mE{sub h}, much more accurate than these previous estimates. Due to the additional efficiency afforded by the algorithm, the excitation energies (T{sub e}) of eight lowest lying excited states: a{sup 3}Π{sub u}, b{sup 3}Σ{sub g}{sup −}, A{sup 1}Π{sub u}, c{sup 3}Σ{submore »

- Authors:

- Department of Chemistry, Frick Laboratory, Princeton University, Princeton, New Jersey 08544 (United States)

- Publication Date:

- OSTI Identifier:
- 22415820

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CLEBSCH-GORDAN COEFFICIENTS; CONFIGURATION INTERACTION; DENSITY MATRIX; DIAGRAMS; DIMERS; ELECTRON CORRELATION; EXCITATION; EXCITED STATES; EXTRAPOLATION; GROUND STATES; HILBERT SPACE; IRREDUCIBLE REPRESENTATIONS; MANY-BODY PROBLEM; MONTE CARLO METHOD; POTENTIAL ENERGY; RENORMALIZATION; SPIN; SU-2 GROUPS; TENSORS

### Citation Formats

```
Sharma, Sandeep, E-mail: sanshar@gmail.com.
```*A general non-Abelian density matrix renormalization group algorithm with application to the C{sub 2} dimer*. United States: N. p., 2015.
Web. doi:10.1063/1.4905237.

```
Sharma, Sandeep, E-mail: sanshar@gmail.com.
```*A general non-Abelian density matrix renormalization group algorithm with application to the C{sub 2} dimer*. United States. doi:10.1063/1.4905237.

```
Sharma, Sandeep, E-mail: sanshar@gmail.com. Wed .
"A general non-Abelian density matrix renormalization group algorithm with application to the C{sub 2} dimer". United States.
doi:10.1063/1.4905237.
```

```
@article{osti_22415820,
```

title = {A general non-Abelian density matrix renormalization group algorithm with application to the C{sub 2} dimer},

author = {Sharma, Sandeep, E-mail: sanshar@gmail.com},

abstractNote = {We extend our previous work [S. Sharma and G. K.-L. Chan, J. Chem. Phys. 136, 124121 (2012)], which described a spin-adapted (SU(2) symmetry) density matrix renormalization group algorithm, to additionally utilize general non-Abelian point group symmetries. A key strength of the present formulation is that the requisite tensor operators are not hard-coded for each symmetry group, but are instead generated on the fly using the appropriate Clebsch-Gordan coefficients. This allows our single implementation to easily enable (or disable) any non-Abelian point group symmetry (including SU(2) spin symmetry). We use our implementation to compute the ground state potential energy curve of the C{sub 2} dimer in the cc-pVQZ basis set (with a frozen-core), corresponding to a Hilbert space dimension of 10{sup 12} many-body states. While our calculated energy lies within the 0.3 mE{sub h} error bound of previous initiator full configuration interaction quantum Monte Carlo and correlation energy extrapolation by intrinsic scaling calculations, our estimated residual error is only 0.01 mE{sub h}, much more accurate than these previous estimates. Due to the additional efficiency afforded by the algorithm, the excitation energies (T{sub e}) of eight lowest lying excited states: a{sup 3}Π{sub u}, b{sup 3}Σ{sub g}{sup −}, A{sup 1}Π{sub u}, c{sup 3}Σ{sub u}{sup +}, B{sup 1}Δ{sub g}, B{sup ′1}Σ{sub g}{sup +}, d{sup 3}Π{sub g}, and C{sup 1}Π{sub g} are calculated, which agree with experimentally derived values to better than 0.06 eV. In addition, we also compute the potential energy curves of twelve states: the three lowest levels for each of the irreducible representations {sup 1}Σ{sub g}{sup +}, {sup 1}Σ{sub u}{sup +}, {sup 1}Σ{sub g}{sup −}, and {sup 1}Σ{sub u}{sup −}, to an estimated accuracy of 0.1 mE{sub h} of the exact result in this basis.},

doi = {10.1063/1.4905237},

journal = {Journal of Chemical Physics},

number = 2,

volume = 142,

place = {United States},

year = {Wed Jan 14 00:00:00 EST 2015},

month = {Wed Jan 14 00:00:00 EST 2015}

}