Werner Kutzelnigg

Theory of magnetic susceptibilities and NMR chemical shifts in terms of localized quantities. II. Application to some simple molecules

A coupled Hartree–Fock method with individual gauge for localized orbitals (IGLO) proposed previously is reformulated and applied to the calculation of the magnetic susceptibility χ and the chemical NMR‐shifts σ of various small molecules. The agreement with experiment is usually very good. Unlike in traditional methods, the results are not very sensitive to the size of the basis, and the application to large molecules does not pose serious problems. The results are analyzed in terms of orbital contributions, and it is shown that local diamagnetic terms are transferable, while local paramagnetic as well as nonlocal contributions are not. Pictorial explanations of the large antishielding effects in F2 and H2CO are given. In CH+3, a large dependence of χ and σ on changes of the geometry (pyramidalization) is found.

Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory

The matrix elements needed in a CI‐SD, CEPA, MP2, or MP3 calculation with linear r12‐dependent terms for closed‐shell states are derived, both exactly and in a consistent approximate way. The standard approximation B guarantees that in the atomic case the error due to truncation of the basis at some angular momentum quantum number L goes as ∼L−7, at variance with L−3 in conventional calculations (without r12 terms). Another standard approximation A has errors ∼L−5, but is simpler and—for moderate basis sets—somewhat better balanced. The explicit expressions for Mo/ller–Plesset perturbation theory of second and third order with linear r12 terms (MP2‐R12 and MP3‐R12, respectively) are explicitly given in the two standard approximations.

r12-Dependent terms in the wave function as closed sums of partial wave amplitudes for large l

The ansatz Ψ= (1+1/2r12)Φ+χ with Φ the bare nuclear (or screened nuclear) wave function and χ expanded in products of one-electron functions is explored for second-order perturbation theory and for variational calculations of the ground state of Helium-like ions.The energy increments El(2)corresponding to the partial wave expansion of χ go asymptotically as l−8, while conventional partial wave increments go as l−4. χ is coupled to Φ by a “residual” interaction U12 that has no singularity for r12=0. With the present ansatz it is sufficient to include l-values up to 5 in order to get the second-order energy accurate to one microhartree. For the same accuracy l≤4 is sufficient in a “CI with correlated reference function” while in conventional CI one must go to l∼50. The surprisingly faster convergence of the variational approach as compared to second-order perturbation theory is explained. The slow convergence of the traditional partial wave expansion is entirely due to the attempt to represent the quantity 1=〈Φ¦r12r12−1¦Φ〉 by its partial wave expansion. The best reference function Φ shows very little shielding and resembles closely the eigenstate of the bare nuclear Hamiltonian. The generalization to arbitrary systems is discussed and it is pointed out that the calculation of “difficult” integrals can be avoided without a significant loss in accuracy.

Rates of convergence of the partial‐wave expansions of atomic correlation energies

The coefficients of the leading terms of the partial‐wave expansion of atomic correlation energies in powers of (l+1/2)−1 are derived for the second‐ and third‐order perturbed energies in the 1/Z expansion for all possible states of two‐electron atoms, and for second‐order Mo/ller–Plesset (many‐body perturbation) theory for arbitrary n‐electron atoms. The expressions for these coefficients given in Table I involve simple integrals over the zeroth‐order wave functions (for the third order energies first‐order wave functions are also involved). The leading term of E(2) goes as (l+1/2)−4 for natural parity singlet states, as (l+1/2)−6 for triplet states, and as (l+1/2)−8 for unnatural parity singlet states. There are no odd powers of (l+1/2)−1 present in E(2), and the coefficient of the (l+1/2)−6 term for natural parity singlet states of two‐electron systems in the 1/Z expansion is generally −5/4 times the coefficient of the (l+1/2)−4 term. In E(3) there are terms that go as odd powers of (l+1/2)−1; the lead...

Møller-plesset calculations taking care of the correlation CUSP

Abstract Moller-Plesset calculations to second order have been carried out on the ten-electron systems Ne, HF and H 2 O with a new functional, including r 12 -dependent pair correlation functions, which takes care of the correlation cusp. The calculated second-order pair energies are accurate to within a few millihartree in comparison with the estimated exact values. In particular, second-order energies of 384.2, 380.1 and 362.9 m E h , have been obtained for Ne, HF and H 2 O respectively.

PNO–CI (pair natural orbital configuration interaction) and CEPA–PNO (coupled electron pair approximation with pair natural orbitals) calculations of molecular systems. I. Outline of the method for closed‐shell states

The methods of configuration interaction with double substitutions to pair natural orbitals (PNO−CI) and of the coupled electron pair approximation (CEPA) proposed by W. Meyer are improved by combination with a new scheme of the calculation of the pair natural orbitals (PNO) and an efficient iterative scheme for the diagonalization of the CI matrix. The relevant matrix elements for the closed shell case are tabulated, the quantities that are pertinent for an analysis of the correlation energy are defined, and the organization of the computer programs is described.

COUPLED CLUSTER THEORY THAT TAKES CARE OF THE CORRELATION CUSP BY INCLUSION OF LINEAR TERMS IN THE INTERELECTRONIC COORDINATES

CC‐R12—a combination of coupled cluster theory and the R12 method, is presented in which the correlation cusp is treated via inclusion of terms explicitly dependent on the interelectronic distance rij into the exponential expansion of the wave function. A diagrammatic derivation of the CC‐R12 equations within the so‐called ‘‘standard approximation B’’ is given at the level of singles, doubles and triples (CCSDT‐R12). MBPT(4)‐R12 is derived as a byproduct of CCSDT‐R12. Fifth order noniterative corrections are also discussed.

PNO–CI (pair natural orbital configuration interaction) and CEPA–PNO (coupled electron pair approximation with pair natural orbitals) calculations of molecular systems. II. The molecules BeH2, BH, BH3, CH4, CH−3, NH3 (planar and pyramidal), H2O, OH+3, HF and the Ne atom

PNO–CI and CEPA–PNO calculations are performed for the molecules MgH2, AlH, AlH3, SiH4, PH3, H2S, HCl, and the Ar atom. Two types of Gaussian basis sets are used; both sets contain one p‐set on H. The ’’small’’ set includes one d‐set on the heavy atom, the ’’standard’’ basis two d‐sets and one f‐set. Both for MgH2 and Ar, a ’’large’’ and a ’’very large’’ basis are used as well, which contain additional polarization functions. The energy improvement due to the different polarization functions is analyzed. Hartree–Fock limits for the molecular energies are estimated. The computed valence shell correlation energies are analyzed in terms of quantities defined in part I, in particular in terms of the IEPA (independent electron pair) correlation energies eμIEPA and the error ΔEIEPA of the IEPA scheme. Both the valence shell interorbital pair correlation energies and the IEPA error are smaller in absolute value than those of the corresponding first row hydrides, provided that one uses the localized representatio...

Quantum chemistry in Fock space. I. The universal wave and energy operators

The Fock space Hamiltonian H has a simpler structure than its projection Hn to n‐particle Hilbert space. It is therefore recommended to diagonalize H—to the extent that this is possible—before one specifies n. Diagonalization of H is possible if one defines the diagonal and nondiagonal parts of an operator appropriately. It is shown that the diagonalized Fock space Hamiltonian L, called an energy operator, contains all information about the eigenvalues of H in a simply coded form. For a spinfree Hamiltonian, spin can be completely eliminated and all interesting quantities are expressible in terms of spinfree excitation operators (generators of the unitary group). The construction of the wave operator W and the energy operator L is formulated in terms of perturbation theory both in the strictly degenerate and the quasidegenerate version. Three variants are discussed that differ in the normalization of W, namely, a intermediate normalization, b and c unitary normalization with two additional conditions, b: ...

The IGLO-Method: Ab-initio Calculation and Interpretation of NMR Chemical Shifts and Magnetic Susceptibilities

In the first part, the theory of molecules in a magnetic field is reviewed with special attention to chemical shifts. The central role of the current density is pointed out. The gauge origin problem and its solution by means of the IGLO method is explained. Various methods for the calculation of chemical shifts are compared and the IGLO method is presented in detail both as far as computations and interpretation of the result are concerned.