# Higher Energy Derivatives in Hilbert Space Multi-Reference Coupled Cluster Theory : A Constrained Variational Approach

^{*}

## Abstract

**:**

## 1 INTRODUCTION

## 2 SRCC AND MRCC LINEAR RESPONSE THEORY

#### 2.1 Single Reference Coupled Cluster Method

^{T}acting on a determinant $|\mathsf{\Phi}\rangle $

^{−T}followed by projection with hole-particle determinants $|{\mathsf{\Phi}}_{ij\dots}^{ab\dots}\rangle $ (henceforth denoted as $|{\mathsf{\Phi}}_{q}\rangle $ with q denoting collective hole-particle excitations respect to $|\mathsf{\Phi}\rangle $), we get the CC equations.

#### 2.2 Analytic Linear Response for SRCC method.

#### 2.3 Z-vector method and other developments

^{(1)}is required for calculating first derivative of energy E

^{(1)}. This has to be done for every mode of perturbation which is a disadvantage. A step towards eliminating this disadvantage was taken by Bartlett and coworkers [8,5,9]. They make use of a technique based on Dalgarno’s interchange theorem [38] used by Handy and Schaefer for configuration interaction (CI) energy derivatives [39]. The technique, known as Z-vector technique, derives algebraic expressions for reducing the number of linear equations for orbital rotations and cluster amplitudes. By inverting equation (7) and substituting in (6), E

^{(1)}can be reorganized as,

^{t},

^{t}vector. The advantage of such reorganization is, unlike earlier equation (7), equation (13) needs to solved only once. This Z-vector method is in some sense equivalent to (2n + 1) type of rule for non-stationary methods. Further simplifications have been carried out by introducing effective CC density matrices [5,9] much akin to CI derivative developments. The technique of Rice and coworkers [40] has also been applied to reduce the number of AO to MO transformations. First applications of CC analytic derivatives have been reported by Bartlett and coworkers [41] and Scheiner and coworkers [42].

#### 2.4 Method of Undetermined Lagrange Multipliers

^{(1)}in favor of a Z-vector is somewhat cumbersome for higher orders and has been pursued by Salter and coworkers [10]. However, Helgakar, Jorgensen and coworkers [17], have pursued an attractive alternative formulation of CC derivative which automatically incorporates Z-vector technique to all orders. Such a formulation proceeds by constructing a Lagrange functional with undetermined multipliers $\Lambda =\{{\lambda}_{q},{\forall}_{q}\}$ corresponding to CC equations (5) as follows.

^{(1)}obey (2n + 1) rule, the undetermined multipliers λ obey (2n + 2) rule [16].

#### 2.5 Multi-Reference Coupled Cluster (MRCC) Theories.

_{0}, is assumed to approximate the quasi-degenerate target space P, spanned by M exact states of exact Hamiltonian H. Hence, it contains the zeroth-order approximations for all the exact states under consideration. Dynamical correlation is brought in via wave-operator Ω mapping model and target space as,

^{T}, is able to yield exact states for different model spaces connected by T. FSMRCC has been successful in accurate computations of spectroscopic quantities such as ionization potential, electron affinity, excitation energies [46,47].

_{eff}. HSMRCC has been pursued by various researchers and has been found to be useful for description of potential energy surfaces [48,49,50]. Spin adaptated formulations have also been pursued and successfully applied to various chemical systems [51].

#### 2.6 Analytic Linear Response for MRCC Theories

_{eff}, were carried out by Pal and coworkers [53]. Working on FSMRCC response equations for one hole model space, they concluded that only highest sector amplitude derivatives can be eliminated from H

_{eff}

^{(1)}under degenerate diagonal H

_{eff}. Further work on HSMRCC theory has shown that it is not possible to eliminate cluster amplitude derivatives in effective Hamiltonian first derivative with a single Z-vector for all states, although M

^{2}number of independent Z-vectors would be sufficient. Recently, however, the present authors have showed that it is possible to eliminate the cluster amplitude derivatives for a chosen state energy derivative. They have derived detailed equations for HSMRCC Z-vector for a chosen state and developed efficient expressions for HSMRCC energy first derivative by extending the idea of effective CC density matrices [35].

## 3 HSMRCC HIGHER ENERGY DERIVATIVES

#### 3.1 The Lagrangian functional and obtaining linear response

_{i}and the set of all Lagrange multpliers corresponding to HSMRCC cluster amplitude equations (19) by Λ. We further collectively refer the set of quantities T,C

_{i},Λ and the energy of i-th state E

_{i}(the Lagrange multiplier corresponding to biorthogonal conditions (23)), by

**Θ**. The following equations summarize the above abreviations.

_{q}(µ) = 0. Making $\mathcal{J}$ stationary with respect to

**Θ**generates sufficient number of equations to determine these parameters. The above functional is similar to one used by Szalay [34] in context with HSMRCC first derivatives. However, the present functional does not introduce C

_{i}dependence in second term. As a consequence, $\mathcal{J}$ leads to Z-vector equations derived by eliminating the first order response of cluster amplitudes, T

^{(1)}from energy first derivative expression. Although current formulation results in slightly complicated expressions for effective CC density matrices, it will be advantageous as discussed in next section.

^{(1)}with strength parameter g is introduced into the Hamiltonian H as,

**Θ**is determined at all strenghts of perturbation g, the functional $\mathcal{J}$ becomes a function of g, denoted by $\mathcal{J}(g,\mathsf{\Theta})$. Response equations can be obtained in two different approaches. First approach, followed by Helgaker, Jorgensen and coworkers [17] and Bartlett and coworkers [10] is to expand the functional $\mathcal{J}$ and the stationary equations obtained for

**Θ**as a Taylor series in strength parameter g. Terms of same order in g in

**Θ**equations are collected and equated to obtain hierarchical equations for various response quantities

**Θ**

^{(n)}.

**$\mathcal{J}$**

^{(n)}is a functional of quantities $\{{\Theta}^{(m)}m=0,n\}$. All response equations upto a required order n can be derived by making the functionals $\left\{{\mathcal{J}}^{(k)}k=0,n\right\}$ stationary with respect to $\{{\mathsf{\Theta}}^{(m)}m=0,n(m\le k)\}$. This leads to the following equations.

**Θ**and

**Θ**

^{(1)}to obtain expressions upto thrid order energy derivatives, and hence equations (32) need to be solved upto n = 1. It should be noted that both approaches are entirely equivalent and lead to identical equations with a given functional.

#### 3.2 Response response equations upto first order

**Θ**upto first order, we use expression for

**$\mathcal{J}$**along with equation (A-1). Zeroth-Order quantities,

**Θ**, can be obtained by

**Θ**, the above equation gives HSMRCC equations for cluster amplitudes (19) and the eigenvalue equations (20)-(21). In addition it gives equations for Λ as,

_{i}(µ) indicates differentiation with respect to specific cluster amplitude t

_{i}(µ) and τ

_{i}(µ) is the hole-particle excitation operator associated with t

_{i}(µ). The quantity [E

_{q}(η)]

_{τi(µ)}has been referred to CC Jacobian by Jorgensen and coworkers [17] and in this case it is the HSMRCC Jacobian. Diagrammatic representation for the above equation can be easily obtained as outlined in [35]. The above equation is the same as the Z-vector equations derived recently by the present authors [35] using elimination technique of Handy and Schaefer [39].

**Θ**

^{(1)}are necessary for calculating higher order energy derivatives and can be obtained by,

**Θ**

^{(1)}. It should be noted that these equations depend on zeroth-order quantities

**Θ**.

^{(1)}. It not only depends on first derivatives of T and C

_{i}, but also on zeroth order quantity Λ. The equations (36) and (40) reveal the same structure pointed out by Jorgensen and coworkers [17] i.e., T

^{(1)}and Λ

^{(1)}are related by the same HSMRCC Jacobian. The only difference between both equations is in the inhomogenous part.

#### 3.3 Simplified expressions for Energy derivatives

**$\mathcal{J}$**

^{(n)}, denoted by ${\mathcal{J}}_{opt}^{(n)}$, when the stationary values of $\{{\mathsf{\Theta}}^{(m)}m=0,n\}$ are substituted in it. Hence, ${\mathcal{J}}_{opt}^{(n)}$ can be considered as the required energy derivative and treat E

_{i}as another Lagrange multiplier. It has been shown that ${C}_{i}^{(n)}$ and T

^{(n)}obey (2n + 1) rule and Lagrange multipliers Λ

^{(n)}and ${E}_{i}^{(n)}$ obey (2n + 2) rule. However, the expressions obtained by simple substitution as above do not take advantage of the above rules. Hence the expressions must be simplified by explicit application of these rules. This eliminates any unnecessary higher order derivatives present in these expressions. Elimination is carried out by referring to appropriate response equations, including the zeroth-order response equations.

_{i}, Λ and E

_{i}. Terms indicated in {} can be eliminated by application of response equations of appropriate order. This is discussed in following subsections. Elimination of higher derivatives of T from the remaining expressions will be demonstrated seperately.

#### 3.3.1 Simplified expression for ${\mathcal{J}}_{opt}^{(1)}$

_{µi}, (20). The Lagrange multipliers Λ

^{(1)}and ${E}_{i}^{(1)}$ are eliminated by zeroth-order T equation (19), and biorthogonality equation (23), respectively. Since HSMRCC equations (20)-(21) contain E

_{i}, presense of E

_{i}is automatically eliminated.

^{(1)}which is present in the surviving terms of (B-1), requires application of zeroth order Λ equations. For demonstate this, ${E}_{eff}^{\mathrm{\nu \mu}}$

^{(1)}and ${E}_{q}^{(1)}$(η) are expanded to seperate out terms containing T

^{(1)}. All such terms cancel precicely from Λ equations, (34). Hence, simplified expression for ${\mathcal{J}}_{opt}^{(1)}$ is given by,

^{(n)}indicates retention of terms containing {T

^{(m)}m = 0,n}. This expression has been further simplified by making use of state-dependent effective CC density matrices [35].

#### 3.3.2 Simplified expression for ${\mathcal{J}}_{opt}^{(2)}$

^{(1)}and ${C}_{i}^{(1)}$, which are determined by equations (36)-(39). ${C}_{i}^{(2)}$ is eliminated by zeroth-order equations for C

_{i}, (20)-(21). Similarly, Λ

^{(2)}, Λ

^{(1)}and ${E}_{i}^{(2)}$are trivially eliminated by T

^{(1)}equation (36), T equation (19) and biorhogonality condition (23) respectively. Although, ${E}_{i}^{(1)}$ can be easily eliminated from equation (39), we deliberately retain it to further simplify the expression. Since ${C}_{i}^{(1)}$ depends on T

^{(1)}, terms containing both T

^{(1)}and ${C}_{i}^{(1)}$ can be eliminated using ${C}_{i}^{(1)}$ response equations (37)- (38). For this some readjustments are necessary as indicated in (B-2).

^{(2)}present in the remaining terms can be eliminated as done in section 3.3.1, by collecting terms containing T

^{(2)}, and making use of zeroth-order Λ equations. This leads to simplified expression for $\mathcal{J}$

^{(2)}as

#### 3.3.3 Simplified expression for ${\mathcal{J}}_{opt}^{(3)}$

^{(1)}is the only additional equation to be solved in addition to T

^{(1)}and ${C}_{i}^{(1)}$ equations. ${C}_{i}^{(3)}$ and ${C}_{i}^{(2)}$ can be eliminated through C

_{i}and ${C}_{i}^{(1)}$ equations respectively. Similary, Λ

^{(3)}, Λ

^{(2)}, ${E}_{i}^{(3)}$, ${E}_{i}^{(2)}$ can be trivially eliminated by applying appripriate response equations.

^{(3)}and T

^{(2)}can be eliminated by collecting terms containing these quantities from surviving terms. It can be easily seen that such terms arise with correct factor necessary for applying Λ and Λ

^{(1)}equations. While T

^{(3)}cancels from Λ equation (34), cancellation of T

^{(2)}requires extraction of terms containing T

^{(2)}in remaining terms with a factor of 3. After collecting such terms, T

^{(2)}cancels by applying Λ

^{(1)}equation (40). Final simplified expression for ${\mathcal{J}}_{opt}^{(3)}$ can be given as,

## 4 DISCUSSION

**Θ**, is also clear. Derivatives of Lagrange numtipliers Λ and E

_{i}, which are not required by the (2n + 2) rule, cancel from lower order response equations of cluster amplitudes T and biorthogonality equations. Cancellation of higher order derivatives of C

_{i}that are not required by the (2n + 1) rule obeyed by it, occurs again by lower order response equation of its conjugate model space coefficient. Since the Lagrange multiplier E

_{i}appears in these equations, some derivatives of E

_{i}go into these cancellations. It should be noticed that while Lagrange multipliers Λ and Λ

^{(1)}appear in ${\mathcal{J}}_{opt}^{(3)}$, the same does not happen for E

_{i}and ${E}_{i}^{(1)}$. In $\mathcal{J}$

^{(3)}, only ${E}_{i}^{(1)}$ appears, while E

_{i}goes into cancellations involving higher order derivatives of C

_{i}. Cancellation of higher derivatives of T not required by (2n +1) rule, happen through lower order Λ response equations. Finally, for even order derivatives, as exemplified in ${\mathcal{J}}_{opt}^{(2)}$, further simplifications are possible through the use of response equations for C

_{i}, resulting in eliminating the terms where both ${C}_{i}^{(1)}$ and T

^{(1)}are present.

_{i}. As a result, T

^{(n)}and Λ

^{(n)}do not share the same HSMRCC Jacobian, as it happens in SRCC. On the other hand, the functional in this work leads to same HSMRCC Jacobian for T

^{(n)}and Λ

^{(n)}. Although this is a clear advantage, other possibile functionals should be explored. However, it should be noted that all such functional are equivalent in the sense that they provide same energies and derivatives differing only in the form of expressions.

^{(1)}can be eliminated from H

_{eff}

^{(1)}in favor of a single Z-vector. It has been concluded to be not possible for a general case, because of matrix nature of H

_{eff}

^{(1)}. Hence, it is necessary to become state-selective and eliminate T

^{(1)}from the energy deriative of a specific state.

## 5 ACKNOWLEDGEMENTS

## APPENDIX A

**$\mathcal{J}$**

^{(n)}in (30) can be obtained by expanding various quantities on the right hand side of (28) as a Taylor series in perturbation strength in g. In the following, superscript on zeroth-order quantities are dropped.

## APPENDIX B

## References

- Cizek, J. Adv. Quant. Chem.
**1969**, 14, 35. - Bartlett, R.J. Annu. Rev. Phys. Chem.
**1981**, 32, 359. - Paldus, J. Methods in Computational Molecular Physics; Wilson, S., Dierckson, G.H.F., Eds.; NATO ASI series B: Plenum, NY, 1992. [Google Scholar]
- Helgaker, T.; Jorgensen, P. Adv. Quant. Chem.
**1988**, 19, 183. - Bartlett, R.J. Geometrical Derivatives of Energy Surface and Molecular properties; Jorgensen, P., Simons, J., Eds.; Reidel, Dordrecht, 1986. [Google Scholar]
- Oslen, J.; Jorgensen, P. Modern Electronic Structure Theory, Part II; Yarkony, D.R., Ed.; World Scientific: Singapore, 1995. [Google Scholar]
- Epstein, S.T. The Variation Principle in Quantum Chemistry; Academic, NY, 1974. [Google Scholar]
- Adamowicz, L.; Laidig, W.D.; Bartlett, R.J. Int. J. Quant. Chem. Symp.
**1984**, 18, 245. - Fitzgerald, G.; Harrison, R.J.; Bartlett, R.J. J. Chem. Phys.
**1986**, 85, 5143. - Salter, E.A.; Trucks, G.W.; Bartlett, R.J. J. Chem. Phys
**1989**, 90, 1752. Salter, E.A.; Bartlett, R.J. J. Chem. Phys**1989**, 90, 1767. - Bartlett, R.J.; Noga, J. Chem. Phys. Lett.
**1988**, 150, 29. Bartlett, R.J.; Kucharski, S.A.; Noga, J. Chem. Phys. Lett.**1989**, 155, 133. - Pal, S.; Ghose, K.B. Curr. Science
**1992**, 63, 667. - Arponen, J. Ann. Phys
**1983**, 151, 311. - Voorhis, T.V.; Head-Gordon, M. Chem. Phys. Lett.
**2000**, 330, 585. Voorhis, T.V.; Head-Gordon, M. J. Chem. Phys.**2000**, 113, 8873. - Jorgensen, P.; Helgaker, T. J. Chem. Phys.
**1988**, 89, 1560. - Helgaker, T.; Jorgensen, P. Theor. Chim. Acta.
**1989**, 75, 111. - Koch, H; Jensen, H.J.A.; Jorgensen, P.; Helgaker, T.; Scuseria, G.E.; Schaefer III, H.F. J. Chem. Phys
**1990**, 92, 4924. - Koch, H.; Jorgensen, P. J. Chem. Phys.
**1990**, 93, 3333. - Mukherjee, D.; Lindgren, I. Phys. Rep.
**1987**, 151, 93. - Mukherjee, D.; Pal, S. Adv. Quant. Chem.
**1989**, 20, 291. - Durand, P.; Malrieu, J.P. Adv. Chem. Phys
**1987**, 67, 321. - Hurtubise, V.; Freed, K.F. Adv. Chem. Phys
**1993**, 83, 465. - Jezioroski, B.; Monkhorst, H.J.M. Phys. Rev. A.
**1982**, 24, 1668. - Meissner, L.; Kucharski, S.A.; Bartlett, R.J. J. Chem. Phys
**1989**, 91, 6187. Meissner, L.; Bartlett, R.J. J. Chem. Phys**1990**, 92, 561. - Mukhopadhyay, D.; Datta, B.; Mukherjee, D. Chem. Phys. Lett.
**1992**, 197, 236. - Mahapatra, U.S.; Datta, B.; Mukherjee, D. Recent Advances in Coupled-Cluster Methods; Bartlett, R.J., Ed.; World Scientific: Singapore, 1997. [Google Scholar]
- Mahapatra, U.S.; Datta, B.; Bandopadhyay, B.; Mukherjee, D. Adv. Quant. Chem
**1998**, 30, 163. - Stanton, J.F. J. Chem. Phys.
**1993**, 99, 8840. - Stanton, J.F.; Gauss, J. J. Chem. Phys.
**1994**, 100, 4695. Stanton, J.F.; Gauss, J. Theo. Chim. Acta.**1995**, 91, 267. Stanton, J.F.; Gauss, J. J. Chem. Phys.**1994**, 101, 8938. - This point is in clarification to the comments of one of the reviewers.
- Pal, S. Phys. Rev. A.
**1989**, 39, 39. - Pal, S. Int. J. Quant. Chem.
**1992**, 41, 443. - Ajitha, D.; Vaval, N.; Pal, S. J. Chem. Phys.
**1999**, 110, 2316. Ajitha, D.; Pal, S. Chem. Phys. Lett.**1999**, 309, 457. Ajitha, D.; Pal, S. J. Chem. Phys.**2001**, 114, 3380. - Szalay, P. Int. J. Quant. Chem.
**1994**, 55, 152. - Shamasundar, K.R.; Pal, S. J. Chem. Phys.
**2001**, 114, 1981. ibid.**2001**, 115, 1979(E). - Monkhorst, H.J. Int. J. Quant. Chem. Symp.
**1977**, 11, 421. - Jorgensen, P.; Simons, J. J. Chem. Phys.
**1983**, 79, 334. - Dalgarno, A.; Stewart, A.L. Proc. Roy. Soc. Lon. Ser A
**238**, 269. - Handy, N.C.; Schaefer III, H.F. J. Chem. Phys.
**1984**, 81, 5031. - Rice, J.E.; Amos, R.D. Chem. Phys. Lett.
**1985**, 122, 585. - Fitzgerald, G.; Harrison, R.J.; Bartlett, R.J. Chem. Phys. Lett.
**1985**, 117, 433. - Scheiner, A.C.; Scuseria, G.E.; Lee, T.J.; Schaefer III, H.F. J. Chem. Phys
**1987**, 87, 5361. - Vaval, N.; Ghose, K.B.; Pal, S. J. Chem. Phys.
**1994**, 101, 4914. - Kumar, A.B.; Vaval, N.; Pal, S. Chem. Phys. Lett.
**1998**, 295, 189. Vaval, N.; Kumar, A.B.; Pal, S. Int. J. Mol. Sci.**2001**, 2, 89. - Vaval, N.; Pal, S. Phys. Rev. A.
**1996**, 54, 250. - Haque, M.; Kaldor, U. Chem. Phys. Lett.
**1985**, 117, 347. ibid.**1985**, 120, 261. Hughes, S.R.; Kaldor, U. Phys. Rev. A.**1993**, 47, 4705. - Pal, S.; Rittby, M.; Bartlett, R.J.; Sinha, D.; Mukherjee, D. J. Chem. Phys.
**1988**, 88, 4357. ibid. Chem. Phys. Lett.**1987**, 137, 273. Vaval, N.; Pal, S.; Mukherjee, D. Theor. Chim. Acc.**1998**, 99, 100. Vaval, N.; Pal, S. J. Chem. Phys.**1999**, 111, 4051. - Meissner, L.; Jankowski, K.; Wasilewski, J. Int. J. Quant. Chem
**1988**, 34, 535. Balkova, A.; Kucharski, S.A.; Meissner, L.; Bartlett, R.J. Theor. Chim. Acta**1991**, 80, 335. ibid. J. Chem. Phys.**1991**, 95, 4311. Balkova, A.; Kucharski, S.A.; Bartlett, R.J. Chem. Phys. Lett**1991**, 182, 511. Kucharski, S.A.; Bartlett, R.J. J. Chem. Phys**1989**, 95, 8227. Balkova, A.; Bartlett, R.J. Chem. Phys. Lett.**1992**, 193, 364. - Paldus, J.; Piecuch, P.; Jeziorski, B.; Pylypow, L. Recent Progress in Many-Body Theories, Vol.3; Ainsworthy, T.L., Campbell, C.E., Clements, B.E., Krotschek, E., Eds.; Plenum Press: New York, 1992. [Google Scholar] Paldus, J.; Piecuch, P.; Pylypow, L.; Jeziorski, B. Phys. Rev. A.
**1993**, 47, 2738. Piecuch, P.; Toboła, R.; Paldus, J. Chem. Phys. Lett.**1993**, 210, 243. Piecuch, P.; Paldus, J. Phys. Rev. A.**1994**, 49, 3479. Kowalski, K.; Piecuch, P. Phys. Rev. A.**2000**, 61. page number. - Berkovic, S.; Kaldor, U. Chem. Phys. Lett.
**1992**, 199, 42. ibid. J. Chem. Phys.**1993**, 98, 3090. - Jezioroski, B.; Paldus, J. J. Chem. Phys
**1988**, 88, 5673. Piecuch, P.; Paldus, J. Theor. Chim. Acta.**1992**, 83, 69. Piecuch, P.; Paldus, J. J. Chem. Phys.**1994**, 101, 5875. Piecuch, P.; Paldus, J. J. Phys. Chem.**1995**, 99, 15354. - Ajitha, D.; Pal, S. Phys. Rev. A.
**1997**, 56, 2658. [CrossRef] - Ajitha, D.; Pal, S. J. Chem. Phys.
**1999**, 111, 3832. ibid.**1999**, 111, 9892(E). - Pal, S. Theor. Chim. Acta.
**1984**, 66, 151. [CrossRef]

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Shamasundar, K.R.; Pal, S.
Higher Energy Derivatives in Hilbert Space Multi-Reference Coupled Cluster Theory : A Constrained Variational Approach. *Int. J. Mol. Sci.* **2002**, *3*, 710-732.
https://doi.org/10.3390/i3060710

**AMA Style**

Shamasundar KR, Pal S.
Higher Energy Derivatives in Hilbert Space Multi-Reference Coupled Cluster Theory : A Constrained Variational Approach. *International Journal of Molecular Sciences*. 2002; 3(6):710-732.
https://doi.org/10.3390/i3060710

**Chicago/Turabian Style**

Shamasundar, K. R., and Sourav Pal.
2002. "Higher Energy Derivatives in Hilbert Space Multi-Reference Coupled Cluster Theory : A Constrained Variational Approach" *International Journal of Molecular Sciences* 3, no. 6: 710-732.
https://doi.org/10.3390/i3060710