Laplace-transformed multi-reference second-order perturbation theories in the atomic and active molecular orbital basis

In the present article, we show how to formulate the partially contracted n-electron valence second order perturbation theory (NEVPT2) energies in the atomic and active molecular orbital basis by employing the Laplace transformation of orbital-energy denominators (OED). As atomic-orbital (AO) basis functions are inherently localized and the number of active orbitals is comparatively small, our formulation is particularly suited for a linearly-scaling NEVPT2 implementation. Some of the NEVPT2 energy contributions can be formulated completely in the AO basis as single-reference second-order M{\o}ller-Plesset perturbation theory and benefit from sparse active-pseudo density matrices - particularly if the active molecular orbitals are localized only in parts of a molecule. Furthermore, we show that for multi-reference perturbation theories it is particularly challenging to find optimal parameters of the numerical Laplace transformation as the fit range may vary among the 8 different OEDs by many orders of magnitude. Selecting the number of quadrature points for each OED separately according to an accuracy-based criterion allows us to control the errors in the NEVPT2 energies reliably

Multi-reference perturbation theory with Cholesky decomposition for the density matrix renormalization group

We present a second-order N-electron valence state perturbation theory (NEVPT2) based on a density matrix renormalization group (DMRG) reference wave function that exploits a Cholesky decomposition of the two-electron repulsion integrals (CD-DMRG-NEVPT2). With a parameter-free multireference perturbation theory approach at hand, the latter allows us to efficiently describe static and dynamic correlation in large molecular systems. We demonstrate the applicability of CD-DMRG-NEVPT2 for spin-state energetics of spin-crossover complexes involving calculations with more than 1000 atomic basis functions. We first assess in a study of a heme model the accuracy of the strongly- and partially-contracted variant of CD-DMRG-NEVPT2 before embarking on resolving a controversy about the spin ground state of a cobalt tropocoronand complex.Comment: 9 pages, 4 figures, 2 table

Second-Order Self-Consistent-Field Density-Matrix Renormalization Group

We present a matrix-product state (MPS)-based quadratically convergent density-matrix renormalization group self-consistent-field (DMRG-SCF) approach. Following a proposal by Werner and Knowles (JCP 82, 5053, (1985)), our DMRG-SCF algorithm is based on a direct minimization of an energy expression which is correct to second-order with respect to changes in the molecular orbital basis. We exploit a simultaneous optimization of the MPS wave function and molecular orbitals in order to achieve quadratic convergence. In contrast to previously reported (augmented Hessian) Newton-Raphson and super-configuration-interaction algorithms for DMRG-SCF, energy convergence beyond a quadratic scaling is possible in our ansatz. Discarding the set of redundant active-active orbital rotations, the DMRG-SCF energy converges typically within two to four cycles of the self-consistent procedureComment: 40 pages, 5 figures, 3 table

Linear interpolation method in ensemble Kohn-Sham and range-separated density-functional approximations for excited states

Gross-Oliveira-Kohn density functional theory (GOK-DFT) for ensembles is in principle very attractive, but has been hard to use in practice. A novel, practical model based on GOK-DFT for the calculation of electronic excitation energies is discussed. The new model relies on two modifications of GOK-DFT: use of range separation and use of the slope of the linearly-interpolated ensemble energy, rather than orbital energies. The range-separated approach is appealing as it enables the rigorous formulation of a multi-determinant state-averaged DFT method. In the exact theory, the short-range density functional, that complements the long-range wavefunction-based ensemble energy contribution, should vary with the ensemble weights even when the density is held fixed. This weight dependence ensures that the range-separated ensemble energy varies linearly with the ensemble weights. When the (weight-independent) ground-state short-range exchange-correlation functional is used in this context, curvature appears thus leading to an approximate weight-dependent excitation energy. In order to obtain unambiguous approximate excitation energies, we propose to interpolate linearly the ensemble energy between equiensembles. It is shown that such a linear interpolation method (LIM) can be rationalized and that it effectively introduces weight dependence effects. As proof of principle, LIM has been applied to He, Be, H$_2$ in both equilibrium and stretched geometries as well as the stretched HeH$^+$ molecule. Very promising results have been obtained for both single (including charge transfer) and double excitations with spin-independent short-range local and semi-local functionals. Even at the Kohn--Sham ensemble DFT level, that is recovered when the range-separation parameter is set to zero, LIM performs better than standard time-dependent DFT.Comment: 26 pages, 8 figure

Nuclear size effects in rotational spectra: A tale with a twist

International audienceWe report a 4-component relativistic benchmark study of the isotopic field shift in the rotational spectrum of three diatomic molecules: TlI, PbTe and PtSi. A central quantity in the theory is the derivative with respect to internuclear distance of an effective electron density associated with a given nucleus, calculated at the equilibrium distance. The effective density, which is related to the mean electron density within the nuclear volume, is usually replaced with the contact density, that is, the electron density at the origin of the nucleus. Our computational study shows that for the chosen systems this induces errors on the order of 10%, which is not acceptable for high-precision work. On the other hand, the systematic nature of the error suggests that it can be handled by an atom-specific correction factor. Our calibration study reveals that relativistic effects increase the contact density gradient by about an order of magnitude, and that the proper transformation of the associated property operator is mandatory in 1- and 2-component relativistic calculations. Our results show very good agreement with the experimental data presented by Schlembach and Tiemann [Chem. Phys. 68 (1982) 21], but disagree completely with the revised results given by the same group in a later paper [Chem. Phys. 93 (1985) 349]. We have carefully re-derived the relevant formulas and cannot see that the rescaling of results is justified. Curiously previous DFT calculations agree quite well with the revised results for TlI and PbTe, but we demonstrate that this is because the authors inadvertently employed a non-relativistic Hamiltonian, which by chance induces an error of the same magnitude as the suggested scaling. For the PtSi molecule our results for the correction term due to nuclear volume disagree with experiment by a factor five, and we recommend a re-examination of the experimental data