Quasi-degenerate Many-body Perturbation Theory Research Articles

Third-order quasidegenerate many-body perturbation theory calculations for valence state correlation energies of the nitrogen and oxygen atoms and their ions

Third order quasidegenerate many-body perturbation theory (QDMBPT) is used to calculate all [2s, 2p] valence state correlation energies of nitrogen and oxygen for all numbers of valence electrons, Nv. The average deviations from experiment for all energies are 0.31 eV for nitrogen and 0.26 eV for oxygen. Rather than using the conventional QDMBPT treatment in which the NN electron matrix elements of the appropriate effective Hamiltonian (HH) are calculated directly, HH is separated into a sum of individual many-electron operators which are independent of Nv. The effective many-electron integrals for a given species are therefore calculated once and then used to construct HH for all values of Nv. This two step procedure has the special virtue of yielding all valence state correlation energies for all valence ions from a single calculation.

Effective Valence Shell Hamiltonian Calculations on Spin-Orbit Coupling of SiH, SiH+, and SiH2+

Recently the ab initio effective valence shell Hamiltonian method H v has been extended to treat spin-orbit coupling in atoms or molecules. The quasidegenerate many-body perturbation theory based H v method has an advantage of determining the spin-orbit coupling energies of all valence states for both the neutral species and its ions with a similar accuracy from a single computation of the effective spin-orbit coupling operator. The new spin-orbit H v method is applied to calculating the fine structure splittings of the valence states of SiH, SiH + , and SiH 2 + not only to assess the accuracy of the method but also to investigate the spin-orbit interaction of highly excited states of SiH species. The computed spin-orbit splittings for ground states are in good agreement with experiment and the few available ab initio computations. The ordering of fine structure levels of the bound and quasi-bound spin-orbit coupled valence states of SiH and its ions, for which neither experiment nor theory is available, is predicted.

The effective valence shell Hamiltonian for spin-orbit coupling

The size extensive, ab initio effective valence shell Hamiltonian method, which is based on quasidegenerate many-body perturbation theory, has been extended to treat spin-orbit coupling in atoms or molecules. Just as the exact projection of the nonrelativistic Hamiltonian into a prechosen valence space enables deriving the multireference perturbation expansion for the exact effective valence shell Hamiltonian, the addition of the Breit–Pauli spin-orbit operator to the original Hamiltonian (as an extra perturbation) enables the use of quasidegenerate many-body perturbation theory to produce the energy independent effective spin-orbit coupling operator that acts within the prechosen valence space. To assess the accuracy of the proposed method, test calculations are performed for the spin-orbit splittings in the valence states of C, Si, Ge, CH, SiH, and GeH and their positive ions using the one-electron spin-orbit approximation with standard values of the effective nuclear charge. The computed spin-orbit splittings are generally in good agreement with experiment and with the few available ab initio computations. Deviations appear in certain cases where the omitted coupling to Rydberg states is known to be relevant. One advantage of the method is that the spin-orbit coupling energies of all valence states for both the neutral species and its ions are simultaneously determined with a similar accuracy from a single computation of the effective spin-orbit coupling operator. Thus, fine structure splittings are predicted for a number of states of each system for which neither experiment nor theory is available. Another advantage stems from the fact that all off-diagonal spin-orbit matrix elements are also obtained.

Valence electronic states of SiH2+ by ab initio effective valence shell Hamiltonian

The ab initio second order effective valence shell Hamiltonian which is based on quasidegenerate many-body perturbation theory has been applied to the SiH2+ dication. From the characteristic properties of Hv, all the valence states are determined with a same accuracy. The four low lying quasibound states (X 2Σ+, A 2Π, a 4Π, C 2Σ+) are found for the dication. It is verified that the existence of some quasibound states is due to the interaction of an attractive state from ion-neutral pair asymptote with a same symmetry repulsive state from ion–ion pair asymptote. Since the experimental data are scarce, this work provides all theoretical spectroscopic properties of these quasibound states. Also the repulsive valence states are examined.

Dipole and transition moments of SiH, PH and SH by ab initio effective valence shell Hamiltonian method

The ab initio effective valence shell Hamiltonian method, based on quasi-degenerate many-body perturbation theory, is generalized to calculate molecular properties as well as the valence state energies. The procedure requires the evaluation of effective operators for each molecular property. Effective operators are perturbatively expanded in powers of correlation and contain contributions from excitations outside of the multireference valence space. To demonstrate the validity of this method, calculations for dipole moments of and transition moments between several low lying valence states of SiH, SiH +, PH, PH +, SH, and SH + to the lowest nontrivial order in the correlations have been performed and compared with other theoretical calculations.

Ab initio effective Hamiltonian calculations on the valence states of SH and SH +

The second-order ab initio effective valence shell Hamiltonian of quasi-degenerate many-body perturbation theory has been used to determine the valence state potential energy curves of SH and SH +. The calculated spectroscopic constants of various valence states are in good agreement with those of experiments or configuration interaction calculations. The experimentally known B 2Σ state has been closely examined. And it is confirmed that the B 2Σ state be the Rydberg 2Σ − state as suggested by Hirst and Guest. This conclusion is readily drawn from the unique nature of the effective valence shell Hamiltonian, which can simultaneously determine the valence states of a molecule and its ions.

Effective valence shell Hamiltonian and potential curves of the oxygen molecule from quasidegenerate many-body perturbation theory

The effective valence shell Hamiltonian (Hv) is calculated for O2 using quasidegenerate many-body perturbation theory with an eight orbital valence space. A comparison is made of the accuracy of Hv results from a second vs third order truncation of the perturbation expansion. Potential curves for ten low lying valence states show that second order calculations produce dissociation energies and harmonic frequencies that are systematically too large. However, the third order Hv calculations correct the deviations present in second order. Our third order ground state spectroscopic constants compared well with those from a full configuration interaction calculation using the same basis set. Hv calculations are also performed using a second set of orbitals constrained such that the molecular valence space is the union of atomic valence spaces. The constrained orbital Hv calculations are designed for comparison with model valence shell Hamiltonians of semiempirical methods. Comparison of second and third order constrained calculations enables a determination of the reliable range of internuclear distances of the individual constrained Hv matrix elements. Third order constrained Hv matrix elements in the atomic orbital basis set are least squares fit to simple functions of inverse internuclear separation or orbital overlap for comparison with the forms used in semiempirical methods. Functional forms employed for second order Hv matrix elements are compared with previous fits to second order Hv matrix elements for S2 and CH in order to present systematic trends.

Comparison of complete model space quasidegenerate many-body perturbation theory for LiH with multireference coupled cluster method

The relative efficacy of using low order trucations with large reference space vs high order methods with small reference space is tested by comparing quasidegenerate many-body perturbation theory (QDMBPT) calculations of potential curves for the five lowest electronic states of LiH with the multireference coupled cluster calculations of Ben-Shlomo and Kaldor [J. Chem. Phys. 89, 956 (1988)]. The infinite order coupled cluster calculations use two configurational reference spaces involving the 2σ, 3σ, and 1π orbitals, while the QDMBPT computations are truncated at either second or third orders and employ the full active reference space formed either from the 2σ, 3σ, and 1π or from the 2σ, 3σ, 4σ, and 1π orbitals. This gives us the opportunity of testing the dependence of QDMBPT computations on the size of reference space, the available freedom in choosing valence orbitals and orbital energies, and the order of truncation. Second order, four valence orbital space QDMBPT calculations provide good agreement with the repulsive portion of the coupled cluster potentials, but yield a separated atom limit that is too high and that therefore distorts the remainder of the potential. Third order improves the separated atom limit considerably, providing good agreement with the coupled cluster calculations. The ‘‘full chemical’’ five orbital reference space, on the other hand, yields very good agreement with coupled cluster potentials when using only the simpler second order QDMBPT calculations, and third order corrections in this case are very small but generally improve agreement with coupled cluster potentials. The five orbital reference space calculations are quite insensitive to a wide range of different choices of valence orbitals and orbital energies, demonstrating a robustness to the QDMBPT formalism used.

Quasidegenerate many-body perturbation theory of CH2

The effective valence shell Hamiltonian (Hν) formulation of quasidegenerate many-body perturbation theory is applied to the CH2 molecule and its positive ion to investigate the ability of the method to describe large spaces having a wide range of orbital energies and patterns of quasidegeneracy that vary greatly with molecular geometry. Sensitive tests are provided by the adiabatic singlet–triplet energy difference, by the lowest singlet excitation energy, by the ionization potentials to a series of ion states, and by the question of whether accurate energies emerge simultaneously for a whole set of valence and ion states from a single Hν computation. Computations assess the dependence of Hν calculations on the choice of orbitals and orbital energies, the only degrees of freedom available to the Hν method once the valence space has been prescribed. Third order corrections are analyzed in preparation for the computations of the following paper that utilize the present calculations as a guide in studying the theoretical basis of the pairwise additivity assumption of semiempirical all valence electron methods.

Dipole moment functions of OH by ab initio effective valence shell hamiltonian method

The ab initio effective valence shell Hamiltonian method, which is based upon quasidegenerate many-body perturbation theory, has been extended to calculate molecular properties. This new method is applied to the study of dipole moment functions of the OH molecule and its ions. The calculated results are in good agreement with those from other approaches. The present calculations demonstrate that the effective valence shell Hamiltonian formalism is also a reliable ab initio method for molecular properties.

Molecular properties by a b i n i t i o quasidegenerate many-body perturbation theory effective Hamiltonian method: Dipole and transition moments of CH and CH+

The ab initio effective valence shell Hamiltonian method, based on quasidegenerate many-body perturbation theory, is generalized to calculate molecular properties as well as the valence state energies which have previously been determined for atoms and small molecules. Our approach is applicable to both expectation values and transition moments of any molecular property within and between the valence states, respectively. The procedure requires the evaluation of effective operators for each molecular property. Effective operators are perturbatively expanded in powers of correlation and contain contributions from excitations outside of the large multireference valence space. Expectation values and transition moments are the diagonal and off-diagonal matrix elements, respectively, of the effective property operators between the eigenfunctions of the correlated effective Hamiltonian. Calculations for dipole moments of and transition moments between several low lying states of CH and CH+ to first order in the correlation corrections are compared with large configuration interaction calculations to show that our methods provide a useful ab initio formalism for dipole moments.

Influence of molecular geometry on valence space for quasidegenerate many-body perturbation theory

The Be-H 2 insertion reaction is used as a model to study the application of quasidegenerate many-body perturbation theory (QDMBPT) to polyatomic molecules where the pattern of quasidegenerate orbitals varies greatly with geometry. Full active valence space QDMBPT calculations are compared with the exact solutions within the basis and with previous QDMBPT computations that retain only a pair of quasidegenerate valence orbitals.

Unitary transformation on the model space. Comparison of Brandow's and Kirtman's versions of quasidegenerate many body perturbation theory

AbstractIn this paper we study the unitary transformations on the model space in quasidegenerate many body perturbation theory (QDMBPT). From this point of view we compare Brandow's and Kirtman's versions of QDMBPT up to the fifth order of perturbation expansion. We show that, starting from Brandow's QDMBPT, we can derive various versions of effective Hamiltonians order by order, of which the Kirtman's version is the simplest one. The operator of unitary transformation we express through the correlation operator.

Electronic structure and bond length dependence of the effective valence shell Hamiltonian of S2 as studied by quasidegenerate many-body perturbation theory

The effective valence shell Hamiltonian (Hv) of S2 is calculated as a function of internuclear distance using quasidegenerate many-body perturbation theory with the full valence space spanned by eight valence orbitals. Calculated potential curves and excitation energies for several valence states are in good agreement with experiment and are compared with configuration interaction calculations using the same primitive basis. In order to test assumptions of semiempirical theories, we also perform a more approximate calculation of Hv in which the valence space is constructed as the union of the atomic valence spaces with the atomic orbitals taken from atomic SCF calculations. A new and important feature of this approximate, correlated Hv is the use of optimized valence and excited orbitals as determined from a constrained SCF procedure. The matrix elements of this approximate, correlated Hv are transformed to the original nonorthogonal atomic valence basis, and their bond length dependences are fit with simple analytical functions. Some calculated Hv matrix elements agree with the forms commonly postulated for semiempirical integrals, while others display quite different behavior. An example of the latter are the one-center, two-electron integrals which depend significantly on bond length in marked contrast to semiempirical theories which assume them to be bond length independent.

Quasidegenerate many-body perturbation theory with self-consistently adapted core reference state (SCA QD MBPT). I. Theoretical formulation and its application to the potential energy curve of the ground state of Li2

The alternative version of Brandow’s QD MBPT is formulated, in which the core vacuum state is redefined with respect to a chosen reference state from model space. This is done through a one-particle operator U in such a way that the total electronic Hamiltonian is not changed. The second order of SCA QD MBPT is applied to the study of the potential energy curve of the ground state of Li2. The results are compared with both the highly sophisticated ab initio calculations (MC SCF, CI-SDT) and the experimental results. The convergency properties of the SCA QD MBPT expansion are discussed with respect to a chosen one-determinantal zeroth-order wave function as well as to an applied model space.

Test of transferability in the exact effective valence shell Hamiltonian of quasi-degenerate many-body perturbation theory: O 2+ calculation from O 2 effective Hamiltonian

Electronic states of the O 2 + ion are evaluated by diagonalization of the exact effective valence shall Hamiltonian H v which has been previously calculated for O 2. The present calculations are then essentially complete valence shall configuration interaction calculations using the same orbitals as for the O 2 calculation, but including correlation contributions from all single and double excitations out of the valence shell. The computed results demonstrate the transferability of the H v of O 2 to the O 2 + system.

General-model-space many-body perturbation theory: The(2s3p)P1,3states in the Be isoelectronic sequence

The energies of the $(2s3p)^{1,3}P$ states of the Be sequence ions N iv-Ne vii are calculated, with use of a general-model-space many-body perturbation theory (MBPT) to third order. The triplet states are described accurately by a one-configuration model ($P$) space including only $2s3p$, but the singlets are not; the singlet-triplet separation has the right sign (singlet below triplet) but is only 30% of experiment. Incorporation of the $2p3s$ configuration in the model space greatly improves the results. This incomplete, two-configuration model space yields virtually the same energies as the larger, complete model space required by other quasidegenerate MBPT methods, at one-fifth the computational cost.

Tests of using large valence spaces in quasidegenerate many-body perturbation theory: Calculations of O2 potential curves

Potential curves are calculated for the oxygen molecule using the effective valence shell Hamiltonian ℋ v method based on quasidegenerate many-body perturbation theory (QDMBPT). Spectroscopic constants of the 12 bound valence states are compared with those obtained from experiment and those calculated by Saxon and Liu from extensive MCSCF and CI calculations. The excellent agreement indicates that the quasidegenerate many-body perturbation scheme is a reliable ab initio method even with larger valence spaces in which the quasidegenerate condition is strongly violated. Approximations to the correlated effective valence shell Hamiltonian full valence shell CI matrix are tested, and new simplifications are found to give results very similar to the complete calculations, thereby providing useful approximation schemes for larger systems. A remarkable property of the QDMBPT-ℋ v method is the fact that all the states at a given internuclear distance are calculated using a common set of core and valence orbitals; the perturbation formulation accounts for the valence orbital readjustments normally required in traditional configuration interaction calculations.

Ab initio determination of bond length dependence of the correlated valence shell Hamiltonian of CH: Comparison with semiempirical theories

The exact ab initio effective valence shell Hamiltonian , which is mimicked by semiempirical theories of valence, is calculated for CH at 11 bond lengths using quasidegenerate many-body perturbation theory to incorporate extensive correlation contributions. Least squares fits of the bond length dependence of the calculated CH matrix elements provide simple formulas which are compared with the intuitive forms introduced into semiempirical theories. Some of the semiempirical formulas, e.g., one-center, one-electron integrals and two-center, two-electron integrals, are in good agreement with our correlated ab initio calculations, while others display substantial departures. For example, the bond length dependence of one-center, two-electron integrals, which are assumed to be independent of bond length in semiempirical theories, is substantial but physically understandable. Corrections are found to the assumed proportionality of resonance and overlap integrals. The bond length dependence of nonclassical three-electron integrals is presented along with the hybrid and exchange integrals that are ignored in zero differential overlap methods.

The correlated pi-Hamiltonian of trans-butadiene as calculated by the ab initio effective valence shell Hamiltonian method: Comparison with semiempirical models

The pi-electron Hamiltonian ℋπ of trans-butadiene is calculated using the effective Hamiltonian quasidegenerate many-body perturbation theory formalism to include ‘‘correlation’’ contributions. When four tight, valencelike orbitals are employed, the calculated ℋπ reproduces experimental and calculated (by configuration interaction) valence state excitation energies, but fails to obtain the diffuse Rydberg-like pi-electron states which interleave the valence spectrum. The addition of a pair of Rydberg molecular orbitals to the valence shell leads to an effective Hamiltonian ℋv which accurately describes both valence and Rydberg states simultaneously. These results provide an explanation as to why semiempirical pi-electron theories have required the use of different parameters, particularly resonance integrals, to calculate different properties. Our calculated ℋπ contains hybrid, exchange, and multicenter two-electron integrals which are customarily ignored to varying degrees in zero differential overlap (ZDO) approximation based methods. The ℋπ integrals often differ considerably from their ‘‘theoretical,’’ uncorrelated counterparts, and our ℋπ calculation provides further insight into these ZDO approximations. A more direct comparison is made with the traditional Pariser–Parr–Pople ℋPPP by averaging our ℋπ to generate a form ℋ̄π similar to ℋPPP. Certain aspects of ℋ̄π are very close quantitatively to ℋPPP, but there are other strong differences which are easily understood physically. The full ℋπ with all repulsion integrals and even effective three-electron interactions can now be used as a testing ground against which new types of semiempirical approximations may be tested.