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State-specific complete active space multireference Møller–Plesset perturbation approach for multireference situations: illustrating the bond breaking in hydrogen halides

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Abstract

Assessment of the complete active space-based state-specific multireference Møller–Plesset perturbation theory, SS-MRMPPT, has been performed on the ground states of HX (X = F, Cl, and Br) systems through the computation of potential energy surface (PES) and spectroscopic constants (such as equilibrium bond lengths, rotational constants, centrifugal distortion constants, vibrational frequencies, anharmonicity constants, and dissociation energies that are closely related to the shape and accuracy of the energy surfaces) extracted from the computed PES. The SS-MRMPPT (involves multiple amplitude sets to parametrize the exact wavefunction) approach isolates one of the several states provided by an effective Hamiltonian in an attempt to avert intruder states in size-extensive manner and hence it forms the basis of a robust approach to the electron correlation problem in cases where a multireference formalism is required. The absence of intruder problem makes SS-MRMPPT an interesting choice for the calculation of the dissociation energy surface(s). The performance of the method has been judged by comparing the results with calculations provided by current generation ab initio methods (multireference or single-reference methods) and we found, in general, a very good accordance between them which clearly demonstrates the usefulness of the SS-MRMPPT method.

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Notes

  1. Of course, the use of genuine multireference methods presents additional levels of complexity for the practitioners, when compared with the corresponding single-reference methods. The genuine multireference approaches often require a degree of subjective judgment from the user to render the calculations manageable and effective. The most obvious conceptual challenge is to choose a meaningful active space for describing a given chemical problem.

  2. Nondynamic correlation is associated with the strongly interacting reference determinants or configurations (via linear combination of the reference functions) while the dynamic one gives contributions to the wave function from the space orthogonal to that spanned by the reference functions, i.e., arising from the couplings between the model and outer spaces. Electronic structure methods capable of providing ‘chemical accuracy’ for ground and electronically excited states of molecules must include both dynamical and nondynamical correlation effects.

  3. Intruder problem is ubiquitous in studying potential energy surfaces leading to the formulation of molecules. In the multireference perturbation theory, intruder state problems (causing appearance of very small energy denominators in PT series and leading to spurious results of the entire PT calculations) are inevitable from the theoretical point of view. On the other hand, for multireference coupled cluster method based on the generalized Bloch equation, intruder problem is a consequence of the polynomial character of amplitude finding equations due to the exponential ansatz for the wave function. Not only that, nonlinear character of the Bloch equation also invite the existence of multiple solutions. Electronic structure methods capable of providing chemical accuracy for ground and excited states of atoms or molecules must be free from such effects. The intruder state problem can usually be corrected by widening the model space, but this inevitably leads to an increase of the computational effort. The main essence of the development of MR-based theory is to employ as small an active or reference space as possible. A well-established workaround for this problem is to focus on one single state of the effective Hamiltonian, leading to state-specific methodology. In most cases, the influence of intruder states becomes more important away from equilibrium regions and successful treatment of this issue will influence the accuracy of predicted energies.

  4. The energy and behavior of a molecule can be expressed as a function of the positions of the nuclei, that is, a potential energy surface and hence many aspects of chemistry can be reduced to questions about potential energy surfaces. A potential energy surface arises naturally when the Born–Oppenheimer approximation is invoked in the solution of the Schrödinger equation for a molecular system.

  5. Traditional multireference Rayleigh–Schrödinger perturbation theory is designed to describe a manifold of states. However, as the perturbation is switched on the relative disposition of these states and those states outside the reference space may change in such a way that convergence of the perturbation series is impaired or even destroyed.

  6. This ansatz was first introduced by Jeziorski and Monkhorst in the context of state-universal multireference coupled cluster.

  7. In SSMR formulation as, via the JM ansatz, every virtual function is generated from each model function by the action of a cluster operator of suitable excitation rank, there is an inherent redundancy of the number of cluster amplitudes vis-a-vis the corresponding configuration interaction coefficients accompanying the virtual function (needed to fully characterize the eigenfunction ψ). Using sufficiency conditions satisfying some important physical requirements (such as the theory be free from intruders and be rigorously size-extensive) Mukherjee et al. developed the SS-MRPT method [28, 29] by invoking a partition of H into H0 (a zeroth-order part) and V (a perturbation), and an order-by-order expansion of cluster operators, T μ of their SS-MRCC formalism [28]. Although sufficiency condition proposed by Mukherjee et al. is very useful, the underlying physical meaning still evades a clear understanding.

  8. In a unrelaxed (frozen) treatment, the updating of the nondynamical correlation as a result of mixing of the virtual functions is not done, assuming approximate additivity of the two effects and thus the methods of frozen coefficients variety (unrelaxed version) may suffer from the internal contraction of the wave function in the reference space. The relaxed (internally decontracted) treatment, on the other hand, dresses the effective operator in the active space, and the diagonalization of this operator automatically relaxes the coefficients of the model functions.

  9. A general difficulty of MRPT is the choice of the zeroth-order Hamiltonian. This is less straightforward than in the SR-based Møller–Plesset perturbation theory, since in the multireference case there is no one-electron Fock operator which is diagonal in the orbital basis. Thus, the zeroth-order Hamiltonian is in general nondiagonal, and a set of linear equations must be solved to determine the first-order wave function. Alternatively, the off-diagonal elements of the zeroth-order Hamiltonian can be neglected, but this may cause additional errors and removes orbital invariance properties.

  10. To the best of our knowledge, these spectroscopic results are the most complete and accurate ones to this day.

  11. See http://www.emsl.pnl.gov/forms/basisform.html.

  12. To assess the comparative performances of electronic structure methods from a perfectly quantitative standpoint, one needs to use the same basis, the same kind of orbitals, and the same geometry. Thereby, one can avoid, or at least attenuate, differences stemming out of the theoretical artifacts while comparing the results. A rigorous comparison of our results with other methods considered here, however, is difficult due to the use of different basis sets. For this reason, the quality of our comparison may not be appropriate. It should be noted that, in this article, our aim is not to look at our method only from the quantitative standpoint. Instead, we attempt to put forth the more qualitative aspect of the method in terms of its predictive power vis-a-vis other standard and established methods in routine use. In view of this, we have also collected the values provided by various methods with different basis and orbitals. To judge our results qualitatively, we also consider the results of various methods with different schemes just as a reference.

  13. Any ab initio calculation inevitably involves both the basis set error (i.e., the error associated with the model employed) and the error of the method itself (i.e., the intrinsic error due to the approximations involved in the method). Thus, when comparing with the experiment, the accuracy of a given post-Hartree–Fock method can only be properly assessed when we can estimate the complete-basis-set limit. This can be accomplished—at least partially—by observing the trend of the computed results while systematically enlarging the basis set.

  14. Although formally simple in conjunction with an explicit intruder-free nature, the BWPT2 method due to Hubač et al. [23–25] is not rigorously size-extensive. In the first applications of BWPT2, no correction to the size-inextensivity was attempted. In a later modification, an attempt was made to expand the target energy in terms of the CASCI energy (by way of expanding the target energy in terms of an unperturbed Rayleigh–Schrödinger-like energy) to get rid of the inextensivity. This has the danger of bringing back the intruders and hence care then has to be exercised to bypass intruders.

  15. In this article, authors study the difference between the various direct perturbation theory and Pauli perturbation method for the HX (X = F,Cl, Br, and I) molecules in order to investigate the relative importance of relativistic effects, higher-order electron correlation effects, and remaining basis sets effects.

  16. To obtain the results at CBS limit, we have used the same scheme as done by Hirata et al. [68].

  17. The results reported in this paper provide very accurate and complete investigations on the molecular parameters of the \(X ^1\Upsigma^{+}\) HBr when compared with the previous theoretical estimations. Their results almost perfectly conform to the available experimental measurements.

  18. In this paper, Peterson et al. performed the D 0, R e , ω e and ω e x e calculations by the CCSD(T) method with a series of correlation-consistent basis sets in conjunction with small-core relativistic pseudopotentials, aug-cc-pVnZ-PP (n = 2, 3, 4, 5). In order to assess the impact of the pseudopotential approximation on the calculated properties, they also made the all-electron CCSD(T) Douglas–Kroll–Hess calculations using the correlation-consistent quintuple basis set augmented with diffused functions, aug-cc-pV5Z-DK.

  19. Also, it has not escaped our attention that to eliminate or considerably reduce the numerical instability of the cluster amplitudes equations of the SS-MRMPT approach, one can use Tikhonov regularization scheme [115], where replaces \(\frac{1}{c_\mu}\) in the coupling term of Eq. (2) by \(\frac{1}{{\widetilde c}_\mu}: \)

    $$ \frac{1}{{\widetilde c}_\mu}= \frac{c_\mu}{c_\mu^2+\tau^2} $$

    where τ is a parameter set by users and is a real quantity. It is evident that \(\frac{1}{{\widetilde c}_\mu}=\frac{1}{c_\mu}\) when c μ has a large value while \(\frac{1}{{\widetilde c}_\mu}=\frac{c_\mu}{\tau^2}\) for very small value of c μ. However, at present we have not incorporated this scheme in our code. We are now engaged in such an implementation which we intend to present in near future. It is worth mentioning at this juncture that the special care for the treatments of the amplitude equation in conjunction with very small value of the reference coefficients are not necessary for the unrelaxed description of the SS-MRMPT method (see [47, 48]).

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Acknowledgments

This paper is dedicated to Professor Shankar Prasad Bhattacharyya, a great teacher, at the occasion of his reaching sixty-five. We have all benefited immensely from our interactions over the years with him. We thank Dr. Debi Banerjee for critical reading of the manuscript. This work has been funded by the Department of Science and Technology of India [Grant No.SR/S1/PC-61/2009]. S.C. acknowledges the infrastructural facility developed in his department through UGC-SAP program.

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  1. Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah, 711103, India

    Sudip Chattopadhyay

  2. Department of Physics, Maulana Azad College, 8 Rafi Ahmed Kidwai Road, Kolkata, 700013, India

    Uttam Sinha Mahapatra

  3. Indian Institute of Astrophysics, Bangalore, 560034, India

    Rajat K. Chaudhuri

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Chattopadhyay, S., Mahapatra, U.S. & Chaudhuri, R.K. State-specific complete active space multireference Møller–Plesset perturbation approach for multireference situations: illustrating the bond breaking in hydrogen halides. Theor Chem Acc 131, 1213 (2012). https://doi.org/10.1007/s00214-012-1213-z

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