Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
.pdf46 |
O. PARISEL AND Y. ELLINGER |
We emphasize that the Rydberg states are included in the variational MCSCF treatment in order not to be undercorrelated relative to the valence states [48,49]. Furthermore, it is seen
that all the previous distributions are coupled to diexcitations from the |
to the |
orbital: that way, we account for the non-dynamical repolarization of the polar |
bond in |
a GVB-like approach. |
|
Finally, we also account for an explicit relaxation of both lone pairs of |
by the |
inclusion of corresponding correlating orbitals [50] and treat them at a GVB-like level.
The final space spanned by these distributions can then roughly be seen as a product of CAS, GVB and MCSCF spaces. It accounts for all effects supposed to be essential to get a good zeroth-order description of the ground and the excited valence and Rydberg states.The dynamical correlation of the 8 electrons included in this variational treatment will be recovered, in the ground state and in the excited states, with the perturbation, and so
will be the correlation energy arising from the core and from the |
remaining electrons. |
The perturbation will also account for any coupling not explicitely included in the
distributions presented in Table 1.
A COUPLED MCSCF-PERTURBATION TREATMENT OF ELECTRONIC SPECTRA |
47 |
4. The vertical electronic spectrum of formaldehyde
4.1. SOME HISTORY ON FORMALDEHYDE STUDIES
Initiated by the pioneering work of Burawoy [51 ], a number of experimental and theoretical studies were performed on the carbonyl group [52-55]. A complete review is beyond the scope of this paper. We will mention only some of them that we consider of particular importance for a comprehensive coverage of the electronic spectrum of formaldehyde for both the theoretical and experimental points of view.
A review of the early experimental works can be found in references [56-58]. More recently, Chutjian recorded the electron-impact excitation spectrum of formaldehyde [59,60] and reported transition energies that are taken as reference values in many other works. So are the experimental values compiled by Robin [61].
A few years ago, Brint et al. [62] focused on the vacuum high-resolution spectrum, pointing out a number of well-defined Rydberg series, of special importance for theoretical benchmarks.
On the theoretical hand, calculations have been performed as soon as in the 50ies [56,63] since formaldehyde represents the smallest member of the carbonyl series. References to early works are avalaible in the compilation by Davidson and McMurchie [64] and in references [56-58,63]. Of particular interest for a comprehensive assignment of the experimental transitions are the very fine and accurate calculations by Harding and Goddard using their GVB-CI method [60,65].
4.2.COMPUTATIONAL DETAILS
The MP2/6-311++G** |
geometry [45] was used for |
in |
the present report |
(CO=1.2122 Å, CH =1.1044 |
Å, HCO=121.94°). It is very close |
to |
the experimental |
geometry [66]. The molecule is supposed to lie in the yz plane; the z axis corresponds to the axis, as in Figure 1.
The MCSCF and the subsequent perturbation calculations were done using a 6-31+G* basis set expanded by a set of spd Rydberg functions. Exponents of this additional gaussians were : 0.032 and 0.028 for the s and p shells for the oxygen atom, and 0.023 and 0.021 for the carbon atom. For the d functions, a common value of 0.015 was chosen for both heavy atoms.
The MCSCF calculation was performed using the configuration space described in section 3.2. The state-averaging was done for seven and seven states for both singlet and triplet multiplicities.
The variational calculations were performed using the Alchemy II package [67] while the further perturbation calculations used a code derived from the original CIPSI module. Proper interfaces between the two programs were developed.
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49 |
4.3.RESULTS AND DISCUSSION
The analysis of the variational wavefunctions clearly shows admixtures of valence and Rydberg characters in many states, either at the orbital level or at the CI level. We will not discuss this point here, but will focuse on transition energies.
The transition energies from the ground state to the lowest 60 vertical excited states considered in this study are reported in Table 2 (30 singlets) and in Table 3 (30 triplets) where they are compared to the avalaible experimental results and to some previous theoretical calculations [45,60,65,68].
It is immediately seen that the agreement of our computed values with experimental transitions is excellent for both valence and Rydberg states. The discrepancies vary from 0.00 eV to 0.40 eV for the largest of them. An exact value of the deviation is however difficult to obtain due to both the experimental band widths and the fact that many observed transitions are not necessary vertical so that structural effects and vibrational shifts are involved. However, the calculated root-mean-square deviation of the computed values from
their experimental assignment is found to be, for the whole spectrum, about |
. To |
our knowledge, there has been no report, whatever might have been the theoretical method used, of such a small deviation between theory and experiment when dealing with so many excited states together.
Within a few exceptions, all singlet states can be correlated to an observed experimental feature. Especially, the high density of states around 11.8 and 12.7 eV is compatible with the observation of unresolved broad peaks in the 11.6-11.9 eV and 12.5-12.8 eV spectral intervals [60]. Unfortunately, the lack of spectroscopic resolution makes any unambiguous one-to-one assignment impossible in these regions.
The situation is more favorable at lower energies: up to about 11 eV, each calculated singlet state correlates unambiguously to a well-resolved experimental line, and the deviation from the experiment does not exceed 0.35 eV which is the largest discrepancy observed. Compared to the calculations by Harding and Goddard [60], the agreement between both methods is excellent. Each state reported by these authors is found in our calculations. In addition, we report some new singlet states of Rydberg character whose description has been made possible essentially because of the larger flexibility of both our MCSCF calculation and one-particle space (basis set including semi-diffuse orbitals that were not in reference [60]). Our calculations provide a clear-cut assignment for the
states which were not reported previously. It is important to notice that most of these new states correlate to the recent experimental results obtained in the study by Brint and Sommer [62] which is devoted to the Rydberg series. It is worth to emphasize that all their lower terms of the ns (3 states), np (6 states) and nd (4 terms) series can be related to a calculated state. Getting a correct description of the higher terms of these series would however require the inclusion of a Rydberg orbital progression in the basis set, so as the consideration of f functions as suggested in reference [62].
The same comments apply to the triplet states, although comparison to experiments is more difficult due to the lack of experimental determinations, even in the low energy region. However, as seen in Table 3, the agreement with avalaible data is excellent, and shows the same quality as for singlet states. So is the correlation with the results by Harding and
Goddard [60]. In the triplet manifold, as in the singlet one, the largest flexibility of the
present method allows for more states to be found: as an example, we tentatively assign the or the state, missing in reference [60], to a peak reported at 9.59 eV [69].
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The comparison to the results obtained using the SCI/MP2 approach [45,68] leads to unquestionable conclusions: not only the SCI/MP2 method does not provide acceptable transition energies for the lowest valence and Rydberg states but it misses some of them and does not provide any good energetical ordering of the excited states. Even if this method presents interesting computational advantages, it can only provide a flimsy quantitative electronic spectrum, as anticipated in section 3.1 and outlined in reference [22].
5. Conclusions and prospects
The present approach is one of the second-generation multireference perturbation treatments first opened by the CIPSI algorithm 20 years ago. Even if the spirit of these new treatments is different, mainly because the reference space is chosen on its completeness rather than on energetical criteria, it remains that the unavoidable problems of disk storage, bottleneck of variational approaches, can now be conveniently transferred to the problem of CPU time which is less restrictive.
The methodology presented here expands the recent CASPT2 approach to more flexible zeroth-order variational spaces for a multireference perturbation, either in the Moller-Plesset scheme or in Epsein-Nesbet approach [70-72]. Furthermore, it allows for the use of a wide set of possible correlated orbitals. These two last points were discussed elsewhere [34].
The reliability of this method for the evaluation of (vertical) electronic spectra has been clearly established in the present work, and further calculations on other molecules (ethylene, vinylydene… for example) have confirmed the very promising potentialities of such an approach that avoids the possible artefacts brought in by any arbitrary truncated CIs when dealing with excited states [49]. We also emphasize that this methodology is able to give reliable splittings between states ranging from 10 kcal/mol to more than 10 eV.
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Reduced Density Matrix versus Wave Function: Recent Developments
C. VALDEMORO
Instituto de Ciencia de Materiales, Serrano 123, 28006 Madrid, Spain
1.Introduction
Much of the great interest that the Reduced Density Matrices (RDM) theory has arisen since the pioneer works of Dirac [1], Husimi [2] and Löwdin [3], is due to the simplification they introduce by averaging out a set of the variables of the many body system under study. For all practical purposes, the averaging with respect to N-1 or N-2 electron variables which is carried out in the 1-RDM or 2-RDM respectively, does not imply any loss of the necessary information. The reason for this is that the operators representing the N-electron observables are sums of operators which depend only on one or two electron variables.
The RDM’s are therefore much simpler objects than the N-electron Wave Function
(WF) which depends on the variables of N electrons. Unfortunately, the search for the N-representability conditions has not been completed and this has hindered the direct use of the RDM’s in Quantum Chemistry. In 1963 A. J. Coleman [4] defined the N-representability conditions as the limitations of an RDM due to the fact that it is derived by contraction from a matrix represented in the N-electron space. In other words, an antisymmetric N-electron WF must exist from which this RDM could have been derived by integrating with respect to a set of electron variables.
The research for finding these conditions, has been intense and fruitful [5-13]. Thus, although an exact procedure for determining directly an N-representable 2-RDM has not been found, many mathematical properties of these matrices are now known and several methods for approximating RDM’s and for employing them have been developed [14-19].
To study the electronic structure of small systems within the framework of the RDM formalism is a good strategy to adopt, but where it is of the foremost importance is in the study of the electronic structure of very large systems. In this latter case, to work within the framework of an N-electron WF does not seem the best approach to take even now that large and fast computers are available. It seems clear to me that it would be advantageous to approach the study of these large systems within a theoretical framework having a quantum statistical character. Since the RDM’s are statistical objects their formalism would fit in a natural way in such a framework.
The aim of this paper is to review the work done by our group in this direction in the last ten years. The reader wishing to have a broader outlook of this vast and fascinating field of research is referred to the Proceedings of the A. J. Coleman Symposium on Reduced Density Matrices and Density Functionals [20]. In this book the opening contribution is by A. J. Coleman himself, where he masterly describes the history of the Reduced Density Matrix (RDM) research from 1929 up to 1987.
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Y. Ellinger and M. Defranceschi (eds.). Strategies and Applications in Quantum Chemistry, 55–75.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.