Relativistic and electron‐correlation effects on magnetizabilities investigated by the Douglas‐Kroll‐Hess method and the second‐order Møller‐Plesset perturbation theory

Isotropic and anisotropic magnetizabilities for noble gas atoms and a series of singlet and triplet molecules were calculated using the second‐order Douglas‐Kroll‐Hess (DKH2) Hamiltonian containing the vector potential A and in part using second‐order generalized unrestricted Møller‐Plesset (GUMP2) theory. The DKH2 Hamiltonian was resolved into three parts (spin‐free terms, spin‐dependent terms, and magnetic perturbation terms), and the magnetizabilities were decomposed into diamagnetic and paramagnetic terms to investigate the relativistic and electron‐correlation effects in detail. For Ne, Kr, and Xe, the calculated magnetizabilities approached the experimental values, once relativistic and electron‐correlation effects were included. For the IF molecule, the magnetizability was strongly affected by the spin‐orbit interaction, and the total relativistic contribution amounted to 22%. For group 17, 16, 15, and 14 hydrides, the calculated relativistic effects were small (less than 3%), and trends were observed in relativistic and electron‐correlation effects across groups and periods. The magnetizability anisotropies of triplet molecules were generally larger than those of similar singlet molecules. The so‐called relativistic‐correlation interference for the magnetizabilities computed using the relativistic GUMP2 method can be neglected for the molecules evaluated, with exception of triplet SbH. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Gauge-origin independent calculation of magnetizabilities and rotational g tensors at the coupled-cluster level

An implementation of the gauge-origin independent calculation of magnetizabilities and rotational g tensors at the coupled-cluster (CC) level is presented. The properties of interest are obtained as second derivatives of the energy with respect to the external magnetic field (in the case of the magnetizability) or with respect to magnetic field and rotational angular momentum (in the case of the rotational g tensor), while gauge-origin independence and fast basis-set convergence are ensured by using gauge-including atomic orbitals (London atomic orbitals) as well as their extension to treat rotational perturbations (rotational London atomic orbitals). The implementation within our existing CC analytic second-derivative code is described, focusing on the required modifications concerning integral evaluation and treatment of the unperturbed and perturbed two-particle density matrices. An extensive set of test calculations for LiH and BH (up to the full configuration-interaction limit), for a series of simple hydrides (HF, H(2)O, NH(3), and CH(4)) as well as the more challenging molecules CO, N(2), and O(3) [employing the CC singles and doubles (CCSD) and the CCSD approximation augmented by a perturbative treatment of triple excitations] demonstrates the importance of electron correlation for high-accuracy predictions of magnetizabilities and rotational g tensors.     

Calculations of frequency-dependent molecular magnetizabilities with quasi-relativistic time-dependent generalized unrestricted Hartree-Fock method

The time-dependent generalized unrestricted Hartree-Fock (TDGUHF) method combined with a two-component quasi-relativistic Hamiltonian generated from the Douglas-Kroll-Hess (DKH) transformation was developed to calculate frequency-dependent molecular magnetizabilities, which are the linear response quantity of a molecule to an external magnetic field. By calculating the magnetizabilities of H(2)X (X = O, S, Se, and Te), the noble gases (He, Ne, Ar, Kr, and Xe) and small open shell molecules (CH(2), CH(3), and O(2)), we found that scalar relativistic terms affect mainly the diamagnetic magnetizability and spin-orbit (SO) interaction affects the paramagnetic magnetizability.     

Interatomic magnetizability: a QTAIM-based approach toward deciphering magnetic aromaticity

Interatomic magnetizability provides insight into the extent of electronic current density between two adjacent atomic basins. By studying a number of well-known aromatic, nonaromatic, and antiaromatic molecules, it is demonstrated that interatomic magnetizability (bond magnetizability) not only is able to verify the exact nature of aromaticity/antiaromaticity among different molecules, but also can distinguish the correct aromaticity order among sets of aromatic/antiaromatic molecules. The interatomic magnetizability is a direct measure of the current flux between two adjacent atomic basins and is the first QTAIM-derived index that evaluates aromaticity based on a response property, that is, magnetizability. Bond magnetizability is easy to compute, straightforward to interpret, and can be employed to evaluate the pure π- or σ-orbital contributions to magnetic aromaticity.     

Breakdown of the pseudopotential approximation for magnetizabilities and electric multipole moments: test calculations for Au, AuF, and Sn(n) cluster (n ≤ 20)

The response of the electronic wavefunction to an external electric or magnetic field is widely considered to be a typical valence property and should, therefore, be adequately described by accurately adjusted pseudopotentials, especially if a small-core definition is used within this approximation. In this paper we show for atomic Au and Au(+), as well as for the molecule AuF and tin clusters, that in contrast to the case of the static electric dipole polarizability or the electric dipole moment, core contributions to the static magnetizability are non-negligible, and can therefore lead to erroneous results within the pseudopotential approximation. This error increases with increasing size of the core chosen. For tin clusters, which are of interest in ongoing molecular beam experiments currently carried out by the Darmstadt group, the diamagnetic and paramagnetic isotropic components of the magnetizability tensor almost cancel out and large-core pseudopotentials do not even predict the correct sign for this property due to erroneous results in both the diamagnetic and (more importantly) the paramagnetic terms. Hence, all-electron calculations or pseudopotentials with very small cores are required to adequately predict magnetizabilities for atoms, molecules and the solid state, making it computationally more difficult to obtain this quantity for future investigations in heavy atom containing molecules or clusters. We also demonstrate for this property that all-electron density functional calculations are quite robust and give results close to wavefunction based methods for the atoms and molecules studied here.     

A polarizable continuum model for molecules at diffuse interfaces

In this work we illustrate an extension of the polarizable continuum model to describe solvation effects on molecules at the interface between two fluid phases (liquid/liquid, liquid/vapor). This extension goes beyond the naive picture of the interface as a plane dividing two distinct dielectrics, commonly employed in continuum models. The main feature of the model is the use of a diffuse interface with an electric permittivity depending on the position. This characteristic clearly allows the study of simple interfaces as well as more complex membrane or multilayer structures. Moreover the smooth variation of the permittivity in the diffuse interface, in contrast to the sharp boundary between two regions, overcomes the numerical divergences due to charges placed at the boundary. The implementation of the model relies on the integral equation formalism, which allows one to calculate the reaction field acting on a molecule immersed in a dielectric environment once the proper Green's function is known. In the present case, such a Green's function is obtained numerically, allowing a large flexibility in the choice of the dielectric permittivity profile. The applications have been selected with the aim of illustrating the capabilities of the model; its present limitations are also discussed.     

A direct relativistic four-component multiconfiguration self-consistent-field method for molecules

A new direct relativistic four-component Kramers-restricted multiconfiguration self-consistent-field (KR-MCSCF) code for molecules has been implemented. The program is based upon Kramers-paired spinors and a full implementation of the binary double groups (D(2h)(*) and subgroups). The underlying quaternion algebra for one-electron operators was extended to treat two-electron integrals and density matrices in an efficient and nonredundant way. The iterative procedure is direct with respect to both configurational and spinor variational parameters; this permits the use of large configuration expansions and many basis functions. The relativistic minimum-maximum principle is implemented in a second-order restricted-step optimization algorithm, which provides sharp and well-controlled convergence. This paper focuses on the necessary modifications of nonrelativistic MCSCF methodology to obtain a fully variational KR-MCSCF implementation. The general implementation also allows for the use of molecular integrals from a two-component relativistic Hamiltonian as, for example, the Douglas-Kroll-Hess variants. Several sample applications concern the determination of spectroscopic properties of heavy-element atoms and molecules, demonstrating the influence of spin-orbit coupling in MCSCF approaches to such systems and showing the potential of the new method.     

Analytical optimization of orbital exponents in Gaussian-type functions for molecular systems based on MCSCF and MP2 levels of fully variational molecular orbital method

We have analyzed the basis function series in molecular systems by optimization of orbital exponents in Gaussian-type functions (GTFs) including the electron correlation effects with multiconfiguration self-consistent field (MCSCF) and M?ller?CPlesset second-order perturbation (MP2) methods. First, we have derived and implemented the gradient formulas of MCSCF and MP2 energies with respect to GTF exponent, as well as GTF center and nuclear geometry, based on the fully variational molecular orbital (FVMO) method. Second, we have applied these electron-correlated FVMO methods to H₂, LiH, and hydrocarbon (CH₄, C₂H₆, C₂H₄, and C₂H₂) molecules. We have clearly demonstrated that the optimized exponent values with electron-correlated methods are different from those with simple Hartree?CFock method, since adequate basis functions for adequate virtual orbitals are indispensable to describe the accurate wave function and geometry for electron-correlated calculations.     

Molecular properties from perturbation theory: A Unified treatment of energy derivatives

The definition of a molecular property as a derivative of the electronic energy with respect to one or more applied perturbations is reviewed. The explicit enumeration of terms entering the derivative formulas is performed by considering in turn the various parameter spaces on which the energy and wave function depend. After deriving general expressions for first, second, and third derivatives for different types of perturbation, the parameter spaces involved in MCSCF and CI cases are identified and used to obtain expressions for the first and second derivatives. An example of an MCSCF third derivative is also given. In addition, the various equation systems defining the perturbed wave functions in each order are derived. Some attention is given to the efficient computer implementation of derivative calculations, and the present work is compared with that of other authors.     

On the SCF calculation of excited states: Singlet states in the two-electron problem

The problem of determining SCF wave functions for excited electronic states is examined for singlet states of two-electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree-Fock plus all single excitation wave functions are equivalent to that of Hartree-Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras-Undeim-MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.     

Theoretical basis of the empirical reaction field approximations through continuum model

We derive from exact integral equations of classical electrostatics some approximate expressions of the interaction energy of a point charge distribution with a dielectric medium. We show how they can be used for computing the wave functions of solvated species imbedded in cavities formed by interlocking spheres in a polarizable continuum. We discuss the relation between these formulae and the ones proposed earlier on empirical basis and we especially emphasize the improvement that they bring out.     

A computational study of some electric and magnetic properties of gaseous BF3 and BCl3

We present the results of an extended computational study of the electric and magnetic properties connected to Cotton-Mouton birefringences, on the trifluoro- and trichloroborides in the gas phase. The electric dipole polarizabilities, magnetizabilities, quadrupole moments, and higher-order hypersusceptibilities--expressed as quadratic and cubic frequency-dependent response functions--are computed within Hartree-Fock, density-functional, and coupled-cluster response theories employing singly and doubly augmented correlation-consistent basis sets and London orbitals in the magnetic property calculations. The results, which illustrate the capability of time-dependent density-functional theory for electron-rich systems, are compared with available experimental data. Revised values of both experimentally derived quadrupole moment of BF3, 2.72 +/- 0.15 a.u., and magnetizability anisotropy of BCl3, -0.45 +/- 0.09 a.u., both obtained in birefringence experiments that neglect the effects of higher-order hypersusceptibilities, are presented. In the theoretical limit the traceless quadrupole moments of BF3 and BCl3 are determined to be 3.00 +/- 0.01 and 0.71 +/- 0.01 a.u., respectively.     

Quantum algorithm for obtaining the energy spectrum of molecular systems

Simulating a quantum system is more efficient on a quantum computer than on a classical computer. The time required for solving the Schr?dinger equation to obtain molecular energies has been demonstrated to scale polynomially with system size on a quantum computer, in contrast to the well-known result of exponential scaling on a classical computer. In this paper, we present a quantum algorithm to obtain the energy spectrum of molecular systems based on the multiconfigurational self-consistent field (MCSCF) wave function. By using a MCSCF wave function as the initial guess, the excited states are accessible. Entire potential energy surfaces of molecules can be studied more efficiently than if the simpler Hartree-Fock guess was employed. We show that a small increase of the MCSCF space can dramatically increase the success probability of the quantum algorithm, even in regions of the potential energy surface that are far from the equilibrium geometry. For the treatment of larger systems, a multi-reference configuration interaction approach is suggested. We demonstrate that such an algorithm can be used to obtain the energy spectrum of the water molecule.     

Adding explicit solvent molecules to continuum solvent calculations for the calculation of aqueous acid dissociation constants

Aqueous acid dissociation free energies for a diverse set of 57 monoprotic acids have been calculated using a combination of experimental and calculated gas and liquid-phase free energies. For ionic species, aqueous solvation free energies were calculated using the recently developed SM6 continuum solvation model. This model combines a dielectric continuum with atomic surface tensions to account for bulk solvent effects. For some of the acids studied, a combined approach that involves attaching a single explicit water molecule to the conjugate base (anion), and then surrounding the resulting anion-water cluster by a dielectric continuum, significantly improves the agreement between the calculated pK(a) value and experiment. This suggests that for some anions, particularly those concentrating charge on a single exposed heteroatom, augmenting implicit solvent calculations with a single explicit water molecule is required, and adequate, to account for strong short-range hydrogen bonding interactions between the anion and the solvent. We also demonstrate the effect of adding several explicit waters by calculating the pK(a) of bicarbonate (HCO(3)(-)) using as the conjugate base carbonate (CO(3)(2-)) bound by up to three explicit water molecules.     

A molecule on the surface or inside a spherical nanoparticle: the computation of the interaction energy in terms of the dielectric continuum model

The electrostatic interaction energy between a charged or polar molecule and a spherical polarizable nanoparticle is studied within the advanced dielectric continuum model proposed previously. The molecule can be located either inside or outside the nanoparticle or in the vicinity of its boundary surface. The nanoparticle and its environment are considered as a polarizable medium and described in terms of a nonuniform dielectric continuum approximation with a position-dependent dielectric permittivity function e(r) \varepsilon (r) , where r is the position vector. A special construction of this function accounts for the proper treatment of sophisticated boundary effects. Test computations are performed for a number of sample molecules.     

Density functional theory for efficient ab initio molecular dynamics simulations in solution

We present a density functional for first-principles molecular dynamics simulations that includes the electrostatic effects of a continuous dielectric medium. It allows for numerical simulations of molecules in solution in a model polar solvent. We propose a smooth dielectric model function to model solvation into water and demonstrate its good numerical properties for total energy calculations and constant energy molecular dynamics.