Pauli Hamiltonian Research Articles

Relativistic regular two-component Hamiltonians

It is shown how the regularized two-component relativistic Hamiltonians of Heully et al. and Chang, Pelissier, and Durand can be viewed as arising from a perturbation expansion that unlike the Pauli expansion remains regular even for singular attractive Coulomb potentials. The performance of these approximate Hamiltonians is tested in the framework of the local density approximation and the relation of their spectrum to that of the Dirac Hamiltonian is discussed. The circumstances under which the current approximations are superior to the Pauli Hamiltonian are analyzed. Finally, it shown how the Hamiltonians could be used within the context of conventional Hartree-Fock theory. © 1996 John Wiley & Sons, Inc.

Electron correlation and relativistic effects in the coinage metal compounds. II. Heteronuclear dimers: CuAg, CuAu, and AgAu

Electric properties of heteronuclear dimers of the coinage metals are calculated at the level of the CCSD(T) approximation applied to 38 electrons of the valence and next-to-valence atomic shells. The relativistic effects are accounted for by using the scalar approximation to the Pauli hamiltonian. Both the pure relativistic and mixed relativistic-correlation contributions to energies and electric properties are computed. All calculations have been carried out by using the recently developed first-order polarized basis sets of the coinage metal atoms. In the non-relativistic approximation all studied dimers show only a moderate degree of polarity; the non-relativistic CuAg turns out to be the most polar dimer with the Cu(−)Ag(+) polarity. The relativistic effects considerably reduce the negative value of the CuAg dipole moment, change the sign of the CuAu dipole moment, and make the AgAu molecule the most polar species in the series. Simultaneously, the parallel component of the dipole polarizability shows only a small relativistic contraction. The calculated quasirelativistic interaction potentials have a correct behavior in the vicinity of their minima and give the Re and ωe values in complete agreement with experiment. Much less satisfactory are the dissociation energy data which seem to suffer from the single reference configuration approximation.

Relativistic and correlation effects in CuH, AgH, and AuH: Comparison of various relativistic methods

The effects of relativity on the bond lengths, dissociation energies, and harmonic vibrational frequencies of the 1Σ+ electronic ground states of the group IB hydrides CuH, AgH, and AuH have been evaluated with a variety of ab initio methods. These properties were investigated with moderately-sized basis sets at the self-consistent field Hartree–Fock (SCF-HF) level and with second-order Mo/ller–Plesset (MP2) perturbation theory for electron correlation. Comparisons were made between all-electron results using the nonrelativistic Hamiltonian, perturbation theory (PT-MVD) at first-order with only the one-electron nonfine-structure terms of the Breit–Pauli Hamiltonian, the spin-free Douglas–Kroll (DK) transformed Dirac Hamiltonian and the untransformed Dirac Hamiltonian, and results using two sets of relativistic effective core potentials (RECPs). The expected trends of bond length decrease, dissociation energy increase, and harmonic vibrational frequency increase with both relativity and correlation are found. Both sets of RECPs are shown to give good results, if accompanied by a reasonable basis set. The DK method is demonstrated to be an inexpensive, reliable approximation to the DHF method.

Fermion determinants in static, inhomogeneous magnetic fields.

The renormalized fermionic determinant of QED in 3 + 1 dimensions, $\mbox{det}_{{ren}}$, in a static, unidirectional, inhomogeneous magnetic field with finite flux can be calculated from the massive Euclidean Schwinger model's determinant, $\mbox{det}_{{Sch}}$, in the same field by integrating $\mbox{det}_{{Sch}}$, over the fermion's mass. Since $\mbox{det}_{{ren}}$ for general fields is central to QED, it is desirable to have nonperturbative information on this determinant, even for the restricted magnetic fields considered here. To this end we continue our study of the physically relevant determinant $\mbox{det}_{{Sch}}$. It is shown that the contribution of the massless Schwinger model to $\mbox{det}_{{Sch}}$ is cancelled by a contribution from the massive sector of QED in 1 + 1 dimensions and that zero modes are suppressed in $\mbox{det}_{{Sch}}$. We then calculate $\mbox{det}_{{Sch}}$ analytically in the presence of a finite flux, cylindrical magnetic field. Its behaviour for large flux and small fermion mass suggests that the zero-energy bound states of the two-dimensional Pauli Hamiltonian are the controlling factor in the growth of $\ln \mbox{det}_{{Sch}}$. Evidence is presented that $\mbox{det}_{{Sch}}$ does not converge to the determinant of the massless Schwinger model in the small mass limit for finite, nonzero flux magnetic fields.

The eigenfunctions of the Pauli Hamiltonian for an iron crystal in the ferromagnetic phase

The symmetrized eigenfunctions of the Pauli Hamiltonian with the spin-orbit term included for an electron in an iron-crystal ferromagnetic domain for three directions of the magnetization: [001], [110] and [111] are calculated. Double magnetic space groups, describing the full symmetry of the problem, and their corepresentations were utilized for the first time to calculate these symmetry-adapted eigenfunctions. A discussion concerning possible applications of the presented eigenfunctions to band-structure calculations within the framework of a tight-binding approach is given.

Exact solutions of regular approximate relativistic wave equations for hydrogen-like atoms

Apart from relativistic effects originating from high kinetic energy of an electron in a flat potential, which are treated in first order by the Pauli Hamiltonian, there are relativistic effects even for low-energy electrons if they move in a strong Coulomb potential. The latter effects can be accurately treated already in the zeroth order of an expansion of the Foldy–Wouthuysen transformation, if the expansion is carefully chosen to be nondivergent for r→0 even for Coulomb potentials, as shown by Van Lenthe et al. [J. Chem. Phys. 99, 4597 (1993)] (cf. also Heully et al. [J. Phys. B 19, 2799 (1986)] and Chang et al. [Phys. Scr. 34, 394 (1986)]). In the present paper, it is shown that the solutions of the zeroth order of this two-component regular approximate (ZORA) equation for hydrogen-like atoms are simply scaled solutions of the large component of the Dirac wave function for this problem. The eigenvalues are related in a similar way. As a consequence, it is proven that under some restrictions, the ZORA Hamiltonian is bounded from below for Coulomb-like potentials. Also, an exact result for the first order regular approximate Hamiltonian is given. The method can also be used to obtain exact results for regular approximations of scalar relativistic equations, like the Klein–Gordon equation. The balance between relativistic effects originating from the Coulombic singularity in the potential (typically core penetrating s and p valence electrons in atoms and molecules) and from high kinetic energy (important for high-energy electrons in a flat potential and also for core-avoiding high angular momentum (d, f, and g states in atoms) are discussed.

A theoretical description of the b 3Π→a 3Σ+ transition of NO+

Using multireference configuration interaction methods, the potential energy curves of the ground and several low-lying excited states of the NO+ ion were calculated. We obtain spectroscopic parameters in good agreement with existing experimental data. In order to establish a one-to-one correspondence between the experimentally known term energies of the recently detected b 3Π→a 3Σ+ transition [Huber and Vervloet, J. Mol. Spectrosc. 146, 188 (1991)] and ab initio data, it is necessary to include explicitly spin–orbit and rotational coupling. Spin–orbit matrix elements were evaluated using the microscopic Breit–Pauli Hamiltonian. The off-diagonal coupling matrix elements 〈b 3Π‖HSO‖a 3Σ+〉 and 〈b 3Π‖L‖a 3Σ+〉 are found to depend strongly on the internuclear separation. The calculated vibrationally averaged fine structure parameter of the b 3Π state for v=0 (67.21 cm−1) is found to be in very good agreement with the value determined experimentally (69.699 cm−1).

Lower Bounds for the Spectra of Schrödinger Operators with Magnetic Fields

An explicit representation of lower bounds for the spectra of Schrödinger operators with magnetic fields on σ-compact Riemannian manifolds is given, using the positivity of the Pauli Hamiltonian. This representation is applied to show some asymptotic properties of a stochastic oscillatory integral and the transverse analyticity of the law of a stochastic line integral.

Ground state of a spin 1/2 charged particle in an even-dimensional magnetic field

We investigate the ground state structure of the Schrodinger operator (Pauli Hamiltonian)H with a magnetic fieldb for a spin 1/2 charged particle in ℝ 2d ≅ ℝ2d ≅ ℂ d . We consider the case whereb is given by the complex exterior derivative of a functionW on ℂ d of the form $$b = i(\bar \partial + \partial )(\bar \partial - \partial )$$ W. We find that dim kerH is related to the asymptotic behavior ofW at infinity. More precisely, if there exists a constantC ∈ ℝ such that there exists the nonzero limit lim|z|→∞e w(z) /|z|−C , then dim kerH is equal to the number of all monomialsf ind variables such that the degree off is smaller than |C| -d. In the case whereC ∉ ℤ, under a weaker assumption this conclusion holds. Moreover, we clarify the structure of kerH.

The spin–orbit effect on potential surfaces of NO2 photodissociation

Potential energy surfaces for photodissociation reaction NO2→NO(2Π)+O(3P) have been studied by ab initio calculations. The effect of spin–orbit interaction on the potential surface features was studied near the product region. All the 12 potential surfaces asymptotically correlated to the NO(2Π)+O(3P) limit were obtained by the state-averaged complete-active-space-self-consistent-field (CASSCF) method. The adiabatic potential surfaces including the spin–orbit interaction were constructed using the full Breit–Pauli Hamiltonian. It was found that the lowest two states are attractive, while all the other states are repulsive. Assuming that NO2 undergoes the photodissociation on the ground state surface, we obtained the bending-rotation energy levels along the dissociation coordinate, and the transition state for each bending level was determined. The potential barriers for the vibrationally adiabatic energy curves were consistent with the recent experiments. Using a simplified model based on the infinite order sudden approximation (IOSA) and the Franck–Condon approximation, the product fine structure distribution was estimated, which is in good agreement with the experimental results.

DYNAMICAL SUPERSYMMETRY AND SOLUTIONS FOR PAULI HAMILTONIANS

A charged point particle interacting with a vortex in 2+1 dimensions (or the relative coordinate of two anyons) possesses a dynamical so(2, 1) symmetry that can be exploited to solve for the case when an external harmonic oscillator potential is added. Moreover, if the particle carries spin 1/2, its interaction with the vortex exhibits a dynamical spl*(2, 1) supersymmetry, which can be imported to the cases where a harmonic oscillator or an external magnetic field is present. Using this supersymmetry, the corresponding Pauli Hamiltonians can be solved algebraically.

Electron-correlation and relativistic contributions to atomic dipole polarizabilities: Alkali-metal atoms.

Electric-dipole polarizabilities of K, Rb, Cs, and Fr are calculated in the framework of the quasirelativistic method based on mass-velocity and Darwin terms in the Pauli Hamiltonian. The electron-correlation contribution due to atomic cores is taken into account at two levels of approximation. The next-to-valence-shell contributions follow from the appropriate complete active-space multiconfiguration self-consistent-field calculations while the remaining core correlation effects are evaluated by using a second-order-perturbation method. Both pure relativistic and mixed correlation-relativistic contributions are evaluated. The present nonrelativistic results for K (44.6 A${\mathrm{\r{}}}^{3}$), Rb (60.8 A${\mathrm{\r{}}}^{3}$), Cs (72.8 A${\mathrm{\r{}}}^{3}$), and Fr (81.8 A${\mathrm{\r{}}}^{3}$) are reduced by 0.8 A${\mathrm{\r{}}}^{3}$, 11.3 A${\mathrm{\r{}}}^{3}$, 11.5 A${\mathrm{\r{}}}^{3}$, and 33.5 A${\mathrm{\r{}}}^{3}$, respectively, due to relativistic and correlation-relativistic corrections. The quasirelativistic results for K, Rb, and Cs are in good agreement with experimental data. The predicted dipole polarizability of Fr is 48.3 A${\mathrm{\r{}}}^{3}$. Additionally, polarized basis sets for Cs and Fr for calculations of molecular electric properties have been generated in this study.

Perturbative relativistic calculations for one-electron systems in a Gaussian basis

The “direct perturbation theory” of relativistic corrections (in which singularities of the traditional relativistic perturbation theory are avoided) is applied in a basis of atom-centered Gaussian orbitals to several states of the H atom and the ions H + 2 and HeH 2+. Almost exponential convergence with the basis size is achieved for the non-relativistic energy as well as for the relativistic corrections E 2, E 4, E 6, respectively. An overall 9-figure accuracy of the relativistic energies of the 1σ g and 1σ u states of H + 2 is achieved. The Pauli—Hamiltonian, that only furnishes E 2, is much inferior to direct perturbation theory even for E 2 (at least for states with non-vanishing electron density of the nuclei).

Soluble many-body systems with flux-tube interactions in an arbitrary external magnetic field.

Exact N-body ground states are considered for the planar system of anyonlike objects governed by the Pauli Hamiltonian in the presence of an arbitrary, position-dependent, external magnetic field. Degeneracies and the maximum number of allowed particles in a given ground state are rigorously determined in terms of the net external magnetic flux and the fluxes carried by the particles. Implications for the physics of quantum Hall effects are also discussed.

The dynamical supersymmetry of the point magnetic vortex

It is shown that the Pauli hamiltonian for a spin- 1 2 charged particle interacting with a magnetic point vortex has a dynamical super spl ∗(2, 1) symmetry, which is the Wick-rotated super spl(2, 1) symmetry, on the plane except at the origin where the magnetic vortex is placed. Using this symmetry, the spectrum and the wave functions are constructed.

Holomorphy of the scattering matrix with respect toc −2 for Dirac operators and an explicit treatment of relativistic corrections

We prove holomorphy of the scattering matrix at fixed energy with respect toc−2 for abstract Dirac operators. Relativistic corrections of orderc−2 to the nonrelativistic limit scattering matrix (associated with an abstract Pauli Hamiltonian) are explicitly determined. As applications of our abstract approach we discuss concrete realizations of the Dirac operator in one and three dimensions and explicitly compute relativistic corrections of orderc−2 of the reflection and transmission coefficients in one dimension and of the scattering matrix in three dimensions. Moreover, we give a comparison between our approach and the firstorder relativistic corrections according to Foldy-Wouthuysen scattering theory and show complete agreement of the two methods.

A b i n i t i o study of the three lowest states X 2∑+, 2Π1/2, 2Π3/2, and B 2∑+ of the HeNe+ ion: Potential energy curves, Λ doubling, and predissociation rates of the rotational levels in the 2Π1/2 (v=0) state

Multireference configuration interaction calculations are performed for the X 2∑+, 2Π, and B 2∑+ states in HeNe+ from 2.2 to 100 bohr. The various perturbations of the Born–Oppenheimer states are evaluated: the spin–orbit interaction is calculated up to second order by employing the Breit–Pauli Hamiltonian. The resulting potential curves for the A 2Π1/2 state are compared with the experimental data of Dabrowski and Herzberg (Ref. 1), the curve of the hereto undetected A 2Π3/2 is predicted. The Λ splitting of the A1 2Π3/2 and the A2 2Π1/2 states due to coupling of the electronic motion with the rotation of the nuclear framework is computed by employing a basis of vibronic eigenfunctions. The calculated parameters of p0=−1.407 cm−1 for A2 2Π1/2 agree well with the corresponding data of p0=−1.417 or −1.427 cm−1 derived from measurements (Refs. 1 and 2). The predissociation rates of the rotational levels in the A2 2Π1/2 (v=0) state are also calculated on the basis of the computed ab initio data. All results are compared with the experimental data of Carrington (Ref. 2). The mechanism for the predissociation of the A 2Π1/2 state (in the absence of a curve crossing) is elucidated.

GENERAL RESULTS FOR THE WITTEN INDEX USING THE WIGNER-KIRKWOOD EXPANSION

A systematic ħ-expansion of the regulated Witten index Δ(β) in one and two-dimensional SUSY quantum mechanics reveals that the lowest order ħ-term is nonzero, and all terms to at least the next four orders vanish. In one dimension, this lowest order term yields the well-known exact quantum result for an arbitrary superpotential. For the Pauli Hamiltonian with an arbitrary vector potential in two-dimensions, we find the new result that the semiclassical Δ(β) is β-independent and is equal to the number of magnetic flux lines.

Relativistically corrected Schrödinger equation.

The relativistically corrected Schr\odinger (RS) equation [(${m}^{2}$${c}^{4}$+${p}^{2}$${c}^{2}$${)}^{1/2}$ +V-E]\ensuremath{\Psi}=0 is sometimes used to describe charged fermions, which are subject to strong Zitterbewegung. Since, however, equations of this type describe bosons with weak Zitterbewegung effects, it is mandatory to screen the Coulomb potential V in an r range of order h/mc, and also to add a screened spin-orbit potential. Thereby the singularity problems with Coulomb potentials are avoided. While the Pauli Hamiltonian must not be used in variational or eigenvalue equations, the RS Hamiltonian may be. An effective potential for the RS equation is derived, which yields relativistically corrected energies and wave functions for fermion systems with Coulomb potentials. Accurate numerical solutions for an unshielded single-center Coulomb potential are presented.

Pauli fermions as components of D=2 supersymmetrical quantum mechanics

We consider the Pauli hamiltonian for charged fermions in electromagnetic and scalar potentials as being a component of the D=2 supersymmetrical hamiltonian. The admissible class of electromagnetic potentials is described. For a subclass of such potentials the fermion spectrum consists of pairs of equally splitted levels. The double degeneracy of levels (the sign of hidden supersymmetry) arises for holomorphic superpotentials.