Improvement of Multiconfigurational Wave Functions and Energies by Correlation Energy Functionals Federico Moscardo ´ , Francisco Mun ˜ oz-Fraile, Angel J. Pe ´ rez-Jime ´ nez, Jose ´ M. Pe ´ rez-Jorda ´ , and Emilio San-Fabia ´ n* Departamento de Quı ´mica Fı ´sica, UniVersidad de Alicante, Apartado 99, E-03080 Alicante, Spain ReceiVed: August 21, 1998; In Final Form: October 26, 1998 Several aspects of hybrid methods resulting from combining multiconfigurational (MC) wave functions with correlation energy functionals are discussed, in particular for a family of functionals recently proposed. 1,2 It is found that adding the correlation at the end of a MC calculation instead of including it in the self-consistent procedure constitutes an excellent approximation. MC potential energy curves and spectroscopic constants show a significant improvement after being corrected with correlation energies from this family of functionals. 1. Introduction Reliable potential energy surfaces are crucial for chemical simulations. Since high-quality descriptions such as full-CI wave functions are out of the question for most systems, it would be particularly interesting to develop a hybrid scheme combining qualitatively correct multiconfigurational (MC) wave functions with cheap but accurate approximations to the unaccounted correlation energy. These qualitatively correct wave functions have to be sought beyond the monoconfigurational Hartree-Fock (HF) method, which gives a poor description 3 of bond dissociation. For example, for the H 2 molecule, the restricted HF model (RHF) severely overestimates the dissociation energy, while the unrestricted HF model (UHF) breaks the spin symmetry. A biconfigurational general valence bond 4 (GVB) treatment of the same molecule gives, on the other hand, a rather good description of the bond, including proper dissociation 5 into atomic fragments. Correlation energy is the difference between the exact and HF energies. Density-functional theory 6,7 (DFT) provides ac- curate estimates of the correlation energy at a reasonable cost, through approximations to the correlation energy functional E c - [F], where F is the electron density (for a review on common approximations, see, for example, ref 8). A particularly simple approach is the Hartree-Fock-Kohn-Sham method 7,9-13 (HFKS), where the exact energy is expressed as the HF functional E HF [F] plus the DFT correlation energy functional An appealing feature of the E c [F] is the possibility of imple- menting it within a post-SCF step, 14 because correlation is a small perturbation that can be added afterward in a non-SCF fashion. Practical experience shows that the following ap- proximation is very accurate 12,15-17 so that it is possible to obtain good estimates of the exact energy by adding the correlation energy obtained from the HF density F HF to the HF energy. The main advantage of such a post-SCF procedure is that the functional derivative δE c [F]/δF is not needed. Although quite good for atoms, 18 the HFKS method does not seem to be much of an improvement over the HF method for molecular spectroscopic constants such as equilibrium distances or vibrational frequencies. 19 The reason is, as we have said, the poor behavior of the HF method for bond dissociation. One would expect that the use of MC wave functions instead of the HF determinant would yield better results where E c,MC is, by definition, the difference between the exact and MC energies. Conventional DFT E c [F] functionals cannot be expected to provide a very good approximation to E c,MC because E c,MC is only a fraction of the total correlation energy. Approximations 1,2,20-39 to E c,MC typically depend on quantities such as natural orbitals or reduced density matrices. 40 Often they are expressed as functionals of the reduced density matrix of second order Γ These functionals can be applied indistinctly to mono- or multiconfigurational Γ’s. For example, if Γ HF and Γ MC come, respectively, from HF and MC wave functions, then E c [Γ HF ] is a good approximation to E c[FHF] and E c [Γ MC ] is a good approximation to E c,MC . Recently, in our laboratory, a family of E c [Γ] functionals has been developed. 1,2,38 Results over atomic Γ HF ’s have been encouraging so far. The goal of the present study is twofold. First, we will check this family of functionals over molecular Γ MC ’s. Second, we will check if a post-SCF procedure analogous to eq 2 is also accurate for E c [Γ]. 2. Self-Consistent Calculations for GVB Wave Functions In order to check the accuracy of the post-SCF scheme, we need, as a reference, a self-consistent procedure for GVB wave functions. We briefly outline it next. For simplicity, we will restrict ourselves to the H 2 molecule and to this wave function, * Corresponding author. Email: sanfa@fisic1.ua.es. E exact ) E HF [F] + E c [F] (1) E HF [F] + E c [F] ≈ E HF + E c [F HF ] (2) E exact ≡ E MC + E c,MC (3) E c,MC ≈ E c [Γ] (4) 10900 J. Phys. Chem. A 1998, 102, 10900-10902 10.1021/jp983448j CCC: $15.00 © 1998 American Chemical Society Published on Web 12/05/1998