# Four-Reference State-Specific Brillouin-Wigner Coupled-Cluster Method: Study of the IBr Molecule

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

_{0}and exact energy ${\tilde{E}}_{0}$ it holds

^{eff}is defined as

_{1}(μ) + T

_{2}(μ) with respect to configuration μ as Fermi vacuum, the wave operator Ω

_{0}in the Hilbert space ansatz [26]

_{0}is the Brillouin-Wigner resolvent

^{eff}), while E

_{q}are the unperturbed energies corresponding to Φ

_{q}similarly as in the Rayleigh‑Schrödinger resolvent, and the notation q >M in Eq. (8) means that internal excitations (the ones relating Φ

_{μ}and Φ

_{ν}, μ,ν ≤ M) are excluded.

_{μμ}is Hamiltonian expectation value for reference configuration Φ

_{μ}and H

_{N}(μ) is the Hamiltonian normally ordered with respect to Fermi vacuum Φ

_{μ}. They can be computed by a modified routine for energy in single-reference CCSD [4]. Coupling between the reference configurations is furnished by the off-diagonal elements

_{ν}and Φ

_{μ}are single- or double-excitations with respect to each other. In the former case we can write either ${\mathrm{\Phi}}_{\nu}={({\mathrm{\Phi}}_{\mu})}_{k}^{c}$ or ${\mathrm{\Phi}}_{\nu}={({\mathrm{\Phi}}_{\mu})}_{\overline{k}}^{\overline{c}}$ and the matrix elements are obtained as r.h.s. of the T

_{1}amplitude update equations. In the latter ${\mathrm{\Phi}}_{\nu}={({\mathrm{\Phi}}_{\mu})}_{k\overline{k}}^{c\overline{c}}$ and the matrix element is obtained from the T

_{2}amplitude equations as described in [4]. Presently, our implementation does not support reference configurations which would be mutually more than biexcited.

_{1}amplitudes read

_{2}case we have

_{ij}is antisymmetrization operator acting on the i, j indices. The r.h.s. term in curly brackets can be identified with the r.h.s. of the single reference CCSD equations computed with internal amplitudes set to zero. As described in[6], the amplitudes obtained by this method need to be corrected for retaining size-extensivity. The idea is to make BWCC a posteriori close to its Rayleigh-Schrödinger (RS) version. Since our BW variant of the Bloch equation (7) contains only a denominator of the type $({\tilde{E}}_{0}-{E}_{q})$, it can be cast into the form

_{1}amplitudes the only term responsible for size-inextensivity is $[{\tilde{E}}_{0}-{H}_{\mu \mu}^{eff}]{t}_{i}^{a}$ on the l.h.s of Eq. (11). With the T

_{2}amplitudes it is the $({\tilde{E}}_{0}-{H}_{\mu \mu}^{eff}){({t}_{ij}^{ab}+{t}_{i}^{a}{t}_{j}^{b}-{t}_{j}^{a}{t}_{i}^{b})}_{\mu}$ term on the l.h.s of Eq. (12) and the last term on the r.h.s. of Eq. (12). From the corrected amplitudes we construct a new H

^{eff}matrix, and by its diagonalization we obtain the final energy. Note that the size-inextensive terms cannot be eliminated during iterations, since the only coupling between amplitudes of different reference configurations, provided by ${\tilde{E}}_{0}$, would be lost.

^{eff}matrix elements are calculated and H

^{eff}is diagonalized. The lowest eigenvalue is then used as ${\tilde{E}}_{0}$ and new T

_{1}and T

_{2}amplitudes for all reference configurations are computed according to Eqs. (11) and (12). This procedure is repeated, employing the DIIS convergence acceleration, until amplitudes converge. Subsequently the a posteriori size-extensivity correction is performed in an additional iteration.

## 3. Computational

^{1}Σ+. Since it has zero angular and spin momentum, it is not subject to spin-orbit splitting, however, scalar relativistic effects play an important role. Therefore in our calculations the inner-shell electrons have been treated using recent averaged relativistic effective core potentials (AREP)[27]. We employed two valence basis sets: the basis set supplied together with the AREP of the size (6s6p1d/3s3p1d) for both I and Br labeled A, and basis set augmented by diffuse functions of size (8s8p3d/5s5p3d) for both atoms, labeled B, which is described in [11] and serves mainly for comparison with that paper. For each internuclear geometry, we performed first a CASSCF calculation with two electrons in the HOMO and LUMO active orbitals, i.e. the same active space as in BWCCSD. The canonical CASSCF orbitals were used in the subsequent BWCCSD treatment based on the four spin‑unrestricted reference configurations possible in this active space. Since the BWCCSD calculation is performed in a spin-unrestricted form, three eigenvalues of H

^{eff}correspond to singlet states and one to a triplet. It has been checked that no spin contamination occurs in numerical procedure. The CASSCF was chosen for the description of the reference state because it is size extensive and correctly describes the static correlation.

_{e}values were obtained as difference between the minimum and the dissociation limit of the potential energy curves. Rovibrational levels (cf. Table 2) have been calculated by the program LEVEL [17].

## 4. Results and Discussion

_{e}has decreased from 47.7 kcal/mol (CCSD) to 38.0 kcal/mol (MR BWCCSD), due to the multireference description at large internuclear separation. The equilibrium bond length r

_{e}has increased by approximately 0.01 Å, while the vibrational frequency ω

_{e}has decreased slightly and the anharmonicity ω

_{e}x

_{e}remained almost same. They are also very similar to values obtained by the MR-CISD method (cf. Table 1). The lack of improvement of the multi-reference method for r

_{e}, ω

_{e}and ω

_{e}x

_{e}with respect to single reference CCSD can be explained by a low contribution of the excited references near the equilibrium geometry and therefore effective reduction of BWCCSD to a single reference CCSD, since the values of r

_{e}, ω

_{e}, and ω

_{e}x

_{e}are determined by the shape of the PES curve near equilibrium. Additional inclusion of dynamical correlation by means of connected triples, when implemented, should result in an improvement of the obtained spectroscopic constants.

**Figure 1.**Potential energy curves of the IBr molecule calculated by the CCSD and MR-BWCCSD methods in the basis set A (see text) shown by solid lines. Energy scale is relative to the sum of energies of isolated atoms, which is indicated by the dashed line. Notice the incorrect dissociation limit of the single-reference CCSD.

**Figure 2.**Potential energy curves of the IBr molecule calculated by the CCSD and MR-BWCCSD methods in the basis set B (see text) shown by solid lines. Energy scale is relative to the sum of energies of isolated atoms, which is indicated by the dashed line. Notice the incorrect dissociation limit of the single-reference CCSD.

Method^{a} | Basis set | r_{e}(Å) | ω_{e}(cm^{‑1}) | ω_{e}x_{e}(cm^{‑1}) | D_{e}(kcal mol^{-1}) |

NR-HF [22] | apVDZ | 2.508 | 284 | 18.4 | |

NR-HF [22] | apVTZ | 2.472 | 295 | 25.7 | |

DC-HF[22] | apVDZ | 2.510 | 276 | 8.0 | |

DC-HF [22] | apVTZ | 2.475 | 287 | 14.5 | |

NR-MP2[22] | apVDZ | 2.537 | 264 | 40.2 | |

NR-MP2[22] | apVTZ | 2.483 | 283 | 50.5 | |

DC-MP2 [22] | apVDZ | 2.541 | 246 | 30.9 | |

DC-MP2 [22] | apVTZ | 2.486 | 275 | 40.4 | |

NR-CCSD [22] | apVDZ | 2.552 | 255 | 37.2 | |

NR-CCSD [22] | apVTZ | 2.497 | 275 | 44.0 | |

DC-CCSD [22] | apVDZ | 2.557 | 246 | 28.0 | |

DC-CCSD [22] | apVTZ | 2.501 | 266 | 34.1 | |

NR-CCSD(T) [22] | apVDZ | 2.562 | 247 | 39.5 | |

NR-CCSD(T) [22] | apVTZ | 2.506 | 267 | 47.1 | |

DC-CCSD(T) [22] | apVDZ | 2.568 | 237 | 30.4 | |

DC-CCSD(T) [22] | apVTZ | 2.511 | 258 | 37.4 | |

HF^{b} [23] | (3s3p1d) | 2.489 | 281 | 16.8 | |

KRHF^{b} [23] | (3s3p1d) | 2.499 | 272 | 7.1 | |

MP2^{b} [23] | (3s3p1d) | 2.510 | 267 | 42.6 | |

KRMP2^{b} [23] | (3s3p1d) | 2.522 | 255 | 33.0 | |

MR-CISD^{b}^{,c,e} | B | 2.534 | 249 | 37.9 | |

MR-CISD^{b}^{,d,e} | B | 2.537 | 247 | 39.3 | |

CCSD^{b}^{,e} [11] | B | 2.520 | 253 | 0.87 | 47.7 |

MR-BWCCSD^{b}^{,f} | A | 2.531 | 252 | 0.87 | 36.5 |

MR-BWCCSD^{b}^{,f} | B | 2.534 | 250 | 0.89 | 38.0 |

experiment [30] | 2.469 | 269 | 0.81 | 42.3 |

^{a}NR means nonrelativistic, DC relativistic.

^{b}Using a relativistic effective core potential.

^{c}The same reference configurations as in MR-BWCCSD; without size-extensivity correction.

^{d}The same reference configurations as in MR-BWCCSD; with Davidson correction.

^{e}Based on RHF molecular orbitals.

^{f}Based on CASSCF molecular orbitals.

**Table 2.**Rovibrational energy levels of IBr obtained from potential curve calculated by MR BWCCSD in basis B (cm

^{‑1})

J | |||||||

v | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

0 | 0.00 | 0.11 | 0.32 | 0.65 | 1.08 | 1.62 | 2.27 |

1 | 245.57 | 245.68 | 245.90 | 246.22 | 246.65 | 247.19 | 247.84 |

2 | 489.16 | 489.27 | 489.48 | 489.80 | 490.23 | 490.77 | 491.42 |

3 | 730.66 | 730.77 | 730.98 | 731.30 | 731.73 | 732.27 | 732.91 |

4 | 969.94 | 970.04 | 970.26 | 970.58 | 971.01 | 971.54 | 972.18 |

_{e}, ω

_{e}and ω

_{e}x

_{e}show a very small difference between the two basis sets used. That means that adding extra diffuse functions to the basis set does not have a large effect on the ground state wave function in the equilibrium interatomic distance range. However the D

_{e}value, which is slightly more sensitive, has changed by approximately 1.5 kcal mol

^{-1}towards the experimental value.

## 5. Conclusions

## Acknowledgements

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^{*}Part of this work has been done during this author’s visit at the Quantum Theory Project, University of Florida, 2301 New Physics Building, P.O. Box 118435, Gainesville, FL 32611 8435, U.S.A.^{†}Also at Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University, Hlavova 2030, 12840 Prague 2, Czech Republic.^{‡}Also at Physics Institute, Silesian University, Opava, Czech Republic.

© 2001 by MDPI (http://www.mdpi.org).

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**MDPI and ACS Style**

Pittner, J.; Demel, O.; Čársky, P.; Hubač, I.
Four-Reference State-Specific Brillouin-Wigner Coupled-Cluster Method: Study of the IBr Molecule. *Int. J. Mol. Sci.* **2001**, *2*, 281-290.
https://doi.org/10.3390/i2060281

**AMA Style**

Pittner J, Demel O, Čársky P, Hubač I.
Four-Reference State-Specific Brillouin-Wigner Coupled-Cluster Method: Study of the IBr Molecule. *International Journal of Molecular Sciences*. 2001; 2(6):281-290.
https://doi.org/10.3390/i2060281

**Chicago/Turabian Style**

Pittner, Jiří, Ondřej Demel, Petr Čársky, and Ivan Hubač.
2001. "Four-Reference State-Specific Brillouin-Wigner Coupled-Cluster Method: Study of the IBr Molecule" *International Journal of Molecular Sciences* 2, no. 6: 281-290.
https://doi.org/10.3390/i2060281