Core Hamiltonian Research Articles

The Grassmann interpolation method for spin-unrestricted open-shell systems.

The recently reported Grassmann interpolation (G-Int) method [J. A. Tan and K. U. Lao, J. Chem. Phys. 158, 051101 (2023)] has been extended to spin-unrestricted open-shell systems. In contrast to closed-shell systems, where G-Int has to be performed only once since the α and β density matrices are the same, spin-unrestricted open-shell systems require G-Int to be performed twice-one for the α spin and another for the β spin density matrix. In this work, we tested the performance of G-Int to the carbon monoxide radical cation CO●+ and nickelocene complex, which have the doublet and triple ground states, respectively. We found that the Frobenius normerrors associated with the interpolations for the α and β spin density matrices are comparable for a given molecular geometry. These G-Int density matrices, when used as an initial guess for a self-consistent field (SCF) calculation, outperform the conventional SCF guess schemes, such as the superposition of atomic densities, purified superposition of atomic densities, core Hamiltonian, and generalized Wolfsberg-Helmholtz approximation. Depending on the desired accuracy, these G-Int density matrices can be used to directly evaluate the SCF energy without performing SCF iterations. In addition, the spin-unrestricted G-Int density matrices have been used for the first time to directly calculate the atomic charges using the Mulliken and ChElPG population analysis.

Simultaneous description of wobbling and chiral properties in even–odd triaxial nuclei

A particle-triaxial rigid core Hamiltonian is semi-classically treated. The coupling term corresponds to a particle rigidly coupled to the triaxial core, along a direction that does not belong to any principal plane of the inertia ellipsoid. The equations of motion for the angular momentum components provide a sixth-order algebraic equation for one component and subsequently equations for the other two. Linearizing the equations of motion, a dispersion equation for the wobbling frequency is obtained. The equations of motion are also considered in the reduced space of generalized phase space coordinates. Choosing successively the three axes as quantization axis will lead to analytical solutions for the wobbling frequency, respectively. The same analysis is performed for the chirally transformed Hamiltonian. With an illustrative example one identified wobbling states whose frequencies are mirror image to one another. Changing the total angular momentum I, a pair of twin bands emerged. Note that the present formalism conciliates between the two signatures of triaxial nuclei, i.e., they could coexist for a single nucleus.

Fully numerical Hartree‐Fock and density functional calculations. II. Diatomic molecules

AbstractWe present the implementation of a variational finite element solver in the HelFEM program for benchmark calculations on diatomic systems. A basis set of the form is used, where (μ, ν, φ) are transformed prolate spheroidal coordinates, B n(μ) are finite element shape functions, and are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree‐Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin density approximation (LDA), generalized gradient approximation (GGA), and the meta‐GGA level can be used through an interface with the Libxc library; meta‐GGA and hybrid DFs are not available in other fully numerical diatomic program packages. Finite electric fields are also supported in HelFEM, enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HelFEM and the novel basis set procedure are demonstrated by reproducing the restricted open‐shell HF limit energies of 68 diatomic molecules from the first to the fourth period with excellent agreement with literature values, despite requiring orders of magnitude fewer parameters for the wave function. Then, the electric properties of the BH and N2 molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed for the literature references. Finally, HF, LDA, GGA, and meta‐GGA calculations of the atomization energy of N2 are performed, demonstrating the superb accuracy of the present approach.

Assessment of Initial Guesses for Self-Consistent Field Calculations. Superposition of Atomic Potentials: Simple yet Efficient.

Electronic structure calculations, such as in the Hartree–Fock or Kohn–Sham density functional approach, require an initial guess for the molecular orbitals. The quality of the initial guess has a significant impact on the speed of convergence of the self-consistent field (SCF) procedure. Popular choices for the initial guess include the one-electron guess from the core Hamiltonian, the extended Hückel method, and the superposition of atomic densities (SAD). Here, we discuss alternative guesses obtained from the superposition of atomic potentials (SAP), which is easily implementable even in real-space calculations. We also discuss a variant of SAD which produces guess orbitals by purification of the density matrix that could also be used in real-space calculations, as well as a parameter-free variant of the extended Hückel method, which resembles the SAP method and is easy to implement on top of existing SAD infrastructure. The performance of the core Hamiltonian, the SAD, and the SAP guesses as well as the extended Hückel variant is assessed in nonrelativistic calculations on a data set of 259 molecules ranging from the first to the fourth periods by projecting the guess orbitals onto precomputed, converged SCF solutions in single- to triple-ζ basis sets. It is shown that the proposed SAP guess is the best guess on average. The extended Hückel guess offers a good alternative, with less scatter in accuracy.

A Feynman dispersion correction: a proof of principle for MNDO.

A dispersion correction is introduced and tested for MNDO. The shift in electron density caused by the interaction between oscillating dipoles in the London picture of dispersion is mimicked by adding a small r-7-dependent attractive nucleus-electron potential to the core Hamiltonian. This potential results in a shift in electron density similar to that used by Feynman to explain dispersion. The resulting parameterized self-consistent and inherently multicenter treatment (MNDO-F) gives good results for CHNO compounds that do not exhibit hydrogen bonds, which MNDO cannot reproduce. This "Feynman" dispersion correction is also applicable to Hartree-Fock and density functional theory. Graphical abstract The MNDO-F optimized geometry for a C60-fullerene tetramer in a tetrahedral configuration.

Natural orbitals of the ground state of the two-electron harmonium atom

The radial components of the natural orbitals (NOs) pertaining to the ^1S_+ ground state of the two-electron harmonium atom are found to satisfy homogeneous differential equations at the values of the confinement strength omega at which the respective correlation factors are given by polynomials. Together with the angular momentum l of the NOs, the degrees of these polynomials determine the orders of the differential equations, eigenvalues of which (arising from well-defined boundary conditions) yield the natural amplitudes. In the case of l=0, analysis of these equations uncovers certain properties of the NOs whereas application of a WKB-like approximation produces asymptotic expressions for both the NOs and the corresponding natural amplitudes that hold when the latter are small negative numbers. Extensive numerical calculations reveal that these expressions remain valid for arbitrary values of omega . The approximate s-type NOs, which are remarkably accurate at sufficiently small radial distances and exhibit universal scaling, differ qualitatively from the eigenfunctions of the core Hamiltonian even at the omega rightarrow infty limit of vanishing electron correlation.

Deformation properties of the projected spherical single particle basis

Deformed single particle energies obtained by averaging a particle–core Hamiltonian with a projected spherical basis depend on a deformation parameter and an arbitrary constant defining the canonical transformation relating the collective quadrupole coordinates and momenta with the boson operators. When the mentioned basis describes the single particle motion of either protons or neutrons the parameters involved are isospin dependent. An algorithm for fixing these parameters is formulated and then applied for 194 isotopes covering a good part of the nuclide chart. Relation with the Nilsson deformed basis is pointed out in terms of deformation dependence of the corresponding single particle energies as well as of the nucleon densities and their symmetries. The proposed projected spherical basis provides an efficient tool for the description of spherical and deformed nuclei in a unified fashion.

Fast and accurate 3D tensor calculation of the Fock operator in a general basis

The present paper contributes to the construction of a “black-box” 3D solver for the Hartree–Fock equation by the grid-based tensor-structured methods. It focuses on the calculation of the Galerkin matrices for the Laplace and the nuclear potential operators by tensor operations using the generic set of basis functions with low separation rank, discretized on a fine N×N×N Cartesian grid. We prove the Ch2 error estimate in terms of mesh parameter, h=O(1/N), that allows to gain a guaranteed accuracy of the core Hamiltonian part in the Fock operator as h→0. However, the commonly used problem adapted basis functions have low regularity yielding a considerable increase of the constant C, hence, demanding a rather large grid-size N of about several tens of thousands to ensure the high resolution. Modern tensor-formatted arithmetics of complexity O(N), or even O(logN), practically relaxes the limitations on the grid-size. Our tensor-based approach allows to improve significantly the standard basis sets in quantum chemistry by including simple combinations of Slater-type, local finite element and other basis functions. Numerical experiments for moderate size organic molecules show efficiency and accuracy of grid-based calculations to the core Hamiltonian in the range of grid parameter N3∼1015.

Semiempirical π-electron models for the calculation of MCD B terms for systems with approximate alternant pairing symmetry. MCD of biphenylene

The recent success of the PPP model in predicting MCD B terms for π-electron transitions in conjugated molecules has not been complete. When applied to alternant hydrocarbons, the pairing symmetry contained in the model leads to a prediction of vanishing B terms for all transitions, in disagreement with experimental observations. In order to calculate the MCD for such “soft” chromophores, complete inclusion of overlap between the basis 2p orbitals in π-electron calculations was investigated. The resulting wave functions did not have the perfect pairing symmetry, and the calculated nonzero B terms of benzenoid hydrocarbons were generally in satisfactory agreement with experiment, provided the choice of parameters did not cause large deviation from perfect pairing. The results also provide an empirical justification for a simplified and, strictly speaking, inconsistent π-electron procedure in which non-neighbor matrix elements are kept for the linear momentum operator but set equal to zero for the core Hamiltonian. Experimental MCD and LD spectra of biphenylene, a 4N-electron perimeter, are presented and compared with the theoretical results.

Dispersion coefficients of the excited states of lithium atoms

The dispersion coefficients of a number of the low-lying states of Li are determined for the homonuclear case. The Li wave functions and energies were computed in a frozen core Hamiltonian with a semiempirical polarization potential. Besides computing the dispersion coefficients, the scalar and tensor polarizabilities and oscillator strengths are computed and generally seen to be in good agreement with other accurate calculations.

Higher-orderCndispersion coefficients for the alkali-metal atoms

The van der Waals coefficients, from ${C}_{11}$ through to ${C}_{16}$ resulting from second-, third-, and fourth-order perturbation theory are estimated for the alkali-metal (Li, Na, K, and Rb) atoms. The dispersion coefficients are also computed for all possible combinations of the alkali-metal atoms and hydrogen. The parameters are determined from sum rules after diagonalizing a semiempirical fixed core Hamiltonian in a large basis. Comparisons of the radial dependence of the ${C}_{n}∕{r}^{n}$ potentials give guidance as to the radial regions in which the various higher-order terms can be neglected. It is seen that including terms up to ${C}_{10}∕{r}^{10}$ results in a dispersion interaction that is accurate to better than 1% whenever the inter-nuclear spacing is larger than $20{a}_{0}$. This level of accuracy is mainly achieved due to the fortuitous cancellation between the repulsive $({C}_{11},{C}_{13},{C}_{15})$ and attractive $({C}_{12},{C}_{14},{C}_{16})$ dispersion forces.

Spin–orbit coupling of spin‐frustrated systems

AbstractWe have investigated the effects of spin–orbit (SO) interactions on noncollinear molecular magnetism by combining the classical Dzyaloshinsky–Moriya (DM) model and ab initio generalized spin orbital (GSO) method. We have derived an estimation scheme of the magnetic anisotropy energy (MAE) and the Dzyaloshinsky vector based on the SO first‐order perturbation theory (SOPT1) for GSO Hartree–Fock (GHF) solutions. We found that the fundamental results of GHF‐SOPT1 method can be reproduced by diagonalizing the core Hamiltonian plus SO terms, and that the spin topologies of odd‐ring systems can be determined by the topological indices of the singly occupied molecular orbitals. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Systematic construction of approximate one-matrix functionals for the electron-electron repulsion energy

The Legendre transform of an (approximate) expression for the ground-state energy E0(η,g) of an N-electron system yields the one-matrix functional Vee[Γ(x′,x)] for the electron-electron repulsion energy that is given by the function Vee(n;g) of the occupation numbers n pertaining to Γ(x′,x) and the two-electron repulsion integrals g computed in the basis of the corresponding natural spinorbitals. Extremization of the electronic energy functional, which is a sum of Vee[Γ(x′,x)] and the contraction of Γ(x′,x) with the core Hamiltonian, produces the (approximate) ground-state energy even if E0(η,g) itself is not variational. Thanks to this property, any electron correlation formalism can be reformulated in the language of the density matrix functional theory. Ten conditions that have to be satisfied by Vee(n;g) uncover several characteristics of Vee[Γ(x′,x)]. In particular, when applied in conjunction with the homogeneity property, the condition of volume extensivity imposes stringent constraints upon the possible dependence of Vee(n;g) on g.

Orthogonalization corrections for semiempirical methods

Based on a general discussion of orthogonalization effects, two new one-electron orthogonalization corrections are derived to improve existing semiempirical models at the neglect of diatomic differential overlap level. The first one accounts for valence-shell orthogonalization effects on the resonance integrals, while the second one includes the dominant repulsive core–valence interactions through an effective core potential. The corrections for the resonance integrals consist of three-center terms which incorporate stereodiscriminating properties into the two-center matrix elements of the core Hamiltonian. They provide a better description of conformational properties, which is rationalized qualitatively and demonstrated through numerical calculations on small model systems. The proposed corrections are part of a new general-purpose semiempirical method which will be described elsewhere.

Nonsingular two/one-component relativistic Hamiltonians accurate through arbitrary high order in ?2

A series of nonsingular two-component relativistic Hamiltonians is derived from the Dirac Hamiltonian by first performing the free-particle Foldy–Wouthuysen transformation and then a block-diagonalizing transformation. The latter is defined in terms of operators which can be determined iteratively through arbitrary order in α, leading to transformed Hamiltonians with the two-component block accurate through α2k, k=1, 2, 3,… . These Hamiltonians give relativistic energies which differ from Dirac's energies only in terms higher than α2k. Their relation to other nonsingular methods of relativistic quantum chemistry (the Douglas–Kroll method, the regular Hamiltonian schemes) is discussed. By removing the spin-dependent operators, the derived Hamiltonians can be written in spin-free one-component form. The computational effort involved is essentially the same as in the case of the Douglas–Kroll scheme and amounts to relatively easy modification of the core Hamiltonian. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 225–239, 1997

On some ways of modifying semiempirical quantum chemical methods

The problems of semiempirical quantum chemical calculations of (a) spin densities in paramagnetic organometallics, (b) hydrogen bonds, and (c) bond energies and the structure of transition-metal compounds are discussed. Some modifications of the existing semiempirical quantum chemical method are presented. An extended NDDO approximation has been developed. This scheme includes explicit symmetric orthogonalization of the core Hamiltonian and the use of Hellmann's effective core potential for core-electron interaction. © 1996 John Wiley & Sons, Inc.

Description of odd- A nuclei in the Pt region in the interacting boson-fermion model

Properties of unique parity states in odd-proton ( 77Ir, 79Au) and odd-neutron nuclei ( 78Pt) are investigated in the framework of the interacting boson-fermion approximation model. The core (boson)-particle (fermion) interaction is represented by a quadrupole-quadrupole interaction and an exchange term, which takes into account the effects of the Pauli exclusion principle. The even-even core nucleus is described in terms of the IBA-1 hamiltonian. The change in the properties of the corresponding odd- A nuclei can be interpreted in terms of a transition of the core hamiltonian between the O(6) and SU(3) limiting cases.

Derivation of a formula for the resonance integral for a nonorthogonal basis set

In a self-consistent field calculation, a formula for the off-diagonal matrix elements of the core Hamiltonian is derived for a nonorthogonal basis set by a polyatomic approach. A set of parameters is then introduced for the repulsion integral formula of Mataga-Nishimoto to fit the experimental data. The matrix elements computed for the nonorthogonal basis set in the pi-electron approximation are transformed to those for an orthogonal basis set by the Löwdin symmetrical orthogonalization.

A description of odd neutron-deficient cadmium isotopes within the rotor-plus-particle model using self-consistent core single-particle states

Single-particle wave functions as obtained from self-consistent calculations for the core are used to calculate within the motor-plus-quasiparticle model, some spectroscopic properties of odd neutron-deficient cadmium isotopes. From constrained Hartree-Fock calculations including pairing correlations and performed with the SIII Skyrme effective force, the deformation energy curves of 98, 102, 106, 110Cd have been extracted together with the single-particle states corresponding to the equilibrium deformations. The projection of standard unified model wave functions onto eigen-functions of the core angular momentum has allowed us to introduce exactly in the core Hamiltonian the experimental level sequence of the neighbouring even isotopes. Therefore all parameters enetering the calculations are either valid for the whole chart of nuclides (effective force parameters) or unambiguously deduced from experimental data (core energies and pairing gaps). In spite of such an absence of free adjustable parameters the recently available experimental data concerning 105, 107, 109, 111Cd isotopes are correctly reproduced both for negative parity (as the expected h 11 2 decoupled band) and for positive parity states. In the 105Cd isotope, we have interpreted a set of positive parity levels as a g 7 2 decoupled band. We have also concluded that a prolated-oblate shape coexistence was absent in the low energy spectra of the isotopes considered.

Towards a new approach for the description of decoupled bands

“Decoupled bands” of odd- A nuclei are described by a core-particle coupling model with the following features: (a) the core Hamiltonian is a fourth-order boson Hamiltonian, (b) the extra nucleon is moving in a shell-model orbit and (c) the coupling Hamiltonian contains terms linear and quadratic in the phonons. The core states are approximated by angular momentum projected “coherent” boson states and the parameters of the core Hamiltonian are fitted to the experimental excitation energies of the core system. With one additional parameter in the coupling Hamiltonian a good fit to the spectra of the odd- A nuclei considered is obtained.