We provide computations for quartet ion states, triplet neutral states, and potential curves for the twisting of neutral allene, quantities that have not previously been reported. All necessary formulas for Hamiltonian matrix elements in the table-CI method are presented in computationally amenable form. In contrast to the initial variant of the method, as developed by Buenker, the obtained formulas allow one to avoid intermediate calculations of separate determinantal matrix elements but instead allows calculation of the Hamiltonian matrix elements directly in the basis of configuration state functions CSFs.

The recently suggested variant of a genealogical scheme for constructing CSFs in the context of table-CI is used, which results in reducing the number of contributions to typical matrix elements by approximately a factor of 2.

The time-dependent extension of density-functional theory TDDFT provides a rigorous formalism allowing the treatment of electronic excitations and excited states. However, just as in traditional ground-state density-functional theory DFT , the quality of the results depends upon the approximation used for the unknown exchange-correlation xc functional.

Perdew and Schmidt have described the various functionals developed for ground state DFT in terms of a Jacob's ladder, where the rungs correspond to successive levels of approximation of the xc-functional[1]. However, TDDFT places additional demands on the functional that are not typically satisfied by approximations developed for the ground-state. The simple time-dependent local approximation already gives remarkably good results for many excited states. However other excitations require more accurate treatment of the xc potential.

Our work on this problem will be summarized in terms of a "Jacob's ladder" adopted to the special needs of applied TDDFT. New theoretical methods are reported for obtaining the binding energies of molecules and other chemical aggregates employing the spectral eigenstates and related properties of their atomic constituents. Wave function antisymmetry in the aggregate atomic spectral-product basis is enforced by unitary transformation performed subsequent to formation of the Hamiltonian matrix, greatly simplifying its construction.

Spectral representatives of the individual atomic number-density operators, which can be determined once and for all and tabulated for future use, provide the computational invariants of the development. Calculations of the lowest-lying attractive and repulsive states of the two-electron pair bond H2 as functions of atomic separation illustrate the nature of the formalism and its convergence to values in accord with results obtained employing conventional methods. The results of large basis set ab initio electronic structure calculations using the RCCSD T method are reported for the bond lengths, bond energies, excitation energies, vibrational frequencies, dipole moments and charge distributions for the titled molecules and where possible compared with experiment and previous calculations The striking differences between the Ca and Zn compounds are discussed in terms of their relative ionic character.

The results suggest that fully relativistic ab initio calculations have the capability to reproduce the experimental bond lengths, harmonic vibrational frequencies and fine structure intervals of the XO series with reasonable accuracy. The quest for spectroscopically accurate XO potentials will provide an excellent benchmark for future theoretical methods. A thorough review of the symmetry properties of the spin-orbit coupling operator and its matrix elements is presented. Various consequences of symmetry upon the practical calculations of spin-orbit coupling matrix elements at both point and double group level are illuminated.

A parallelisation scheme and various steps unrelated to symmetry toward making the calculations more efficient are discussed. The potential energy surfaces of these two states are almost degenerate, meaning that the bimodal state distribution observed for the 2A' state cannot be explained by the topography of the adiabatic surfaces.

It is shown that the bimodal distribution originates from concomitant adiabatic and non-adiabatic dissociation processes from the 2A' state. The non-adiabatic transition is due to unlocking of an S atom electron orbital from the C-S axis by rapid rotation of CO. There has been a renewed interest in the properties of the O4 system in recent years coming from a variety of scientific fields high resolution spectroscopy, nature of intermolecular forces, molecular beam scattering, molecular collision dynamics, photochemistry, atmospheric chemistry, solid state physics and aimed at a better understanding of its electronic structure and dynamics.

Several two-state reaction path models of the potential energy surfaces and spin-orbit coupling are studied.

In these models, the triplet state curve shows a barrier along the reaction path and the singlet state a well such that the two states intersect at a location near the barrier top. Eleven choices of the parameters in the Hamiltonian are examined in which the effect of the triplet-singlet crossing location, the singlet well depth, and the size and coordinate dependence of the spin-orbit coupling are varied. The quantum calculations show that if the crossing occurs on the reagent side of the triplet barrier, and the spin-orbit coupling at that point is similar to what exists in the reagent O atom, then the low energy reactivity is dominated by intersystem crossing.

This result is reasonably well described by surface hopping within the diabatic representation; the corresponding adiabatic representation results are less accurate below the adiabatic threshold, but more accurate above threshold.

## Wave energy mcq

The influence of Stuckelberg interference effects on the state-resolved reaction probabilities is also studied. The calculations do capture the large fluctuations of the dissociation rate with respect to the HOCl total angular momentum, and a simple, general model is presented to rationalize these results. Limited, new calculations of HO37Cl dissociation show a dramatic isotope effect on the dissociation rate. In this publication we consider the electronically multi-fold degeneracy with the aim of revealing the connection between the adiabatic-to-diabatic transformation matrices and Wigner's irreducible representation of the rotation group.

To form the connection we constructed simplified models of two, three and four states, all electronically degenerate at a single point, and we employed the relevant non-adiabatic coupling matrices. We found that once these matrices are properly quantized Baer, M.

Far-infrared, Raman, and dispersed fluorescence were utilized to obtain the vibrational data for the S ground states. This allowed the potential energy surfaces PESs of these molecules to be determined in terms of the ring-puckering and ring-flapping coordinates for both states. However, as noted by Peterson and Dunning [42], if the goals of a calculation are to obtain chemical accuracy of thermochemical properties, the effects of correlating the electrons in the core orbital generally must be addressed in the calculations. Of course, ab initio calculations including correlation effects of core electrons are very time-consuming.

In this work, not only appropriate active space but also an optimized basis set with additional functions for describing core and core-valence correlation effects called CV is employed in our ab initio calculations, which can also guarantee that the two lowest-lying singlet PESs are degenerate at linearity.

The organization of the present article is as follows. Section 2 describes the ab initio electronic structure calculations. The fitting of the ab initio energy points is presented in Section 3. Finally, a summary is given in Section 5. Electronic Structure. For computational convenience, the molecule is placed in the yz plane, and the electronic configuration of ground state is 1a1 n 2a1 n 1b2 n 3a1 t 1b1 t.

The 1a1 and 2a1 molecular orbitals MOs mostly have carbon 1s and 2s characters, respectively. The 3a1 , 1b1 , and 1b2 MOs mainly have the carbon 2p character lying along the twofold z axis, perpendicular and parallel with respect to the yz plane, respectively. In the Cs symmetry, the a1 and b2 orbitals become the a' orbitals, and the b1 and a2 orbitals become the a'' orbitals. The X3B1 and b1B1 states, which share the same spatial orbital configuration with two open-shell electrons parallelly or antiparallelly distributed in the outer 3a1 and 1b1 orbitals, correspond to the triplet and singlet configurations, respectively.

While the a1A1 and c1A1 states correspond primarily to configurations with double occupation of the 3a1 or 1b1 orbital, they could be appropriately described by the two main configuration wave functions. Hence, multireference methods are required for an accurate ab initio description of the above excited states. C4 1ai ,t 2ai ,t 1b2 ,t 3ai 0 1bi ,t 11A2: 1ai ,t 2ai ,t 1b2 ' 3ai tt 1bi t 11B2: 1fli ,t 2ai ,t 1b2 , 3ai t 1bi ,t. The spin-orbit coupling between the X3B1 and singlet states is small in the well region and thus is neglected in this work.

Two levels of ab initio calculations were performed for the PES construction which involves a dual-level strategy [43]. The electronic energies in the lower-level calculations are calculated with the state-averaged complete active space self-consistent field SA-CASSCF and internally contracted multireference configuration interaction icMRCI methods [].

The active space consists of six electrons distributed among seven orbitals, which correspond to all valence electrons and valence orbitals and one additional 3s orbital of carbon which is of Rydberg character. Dunning's correlation-consistent polarized valence quadruple-zeta basis set augmented with diffuse functions aug-cc-pVQZ is used. In the higher-level calculations, the methods and algorithm are the same as the lower-level, but the active space and basis set are different.

The active space consists of all electrons distributed among eight orbitals, which include 1s, all valence, and 3s orbital of carbon. The details of this scheme could be found elsewhere [42, ], and only a brief outline will be given here. The seven inner 1s functions are contracted to two functions using the coefficients from the aug-cc-pVQZ basis set. The outer five s functions are uncontracted as the six p functions. Two tight d and f functions are added to the 3d, 2f, 1g polarized set given by Peterson and Dunning [42] and Woon and Dunning [48].

The additional functions are even tempered extensions of the valence sets, and the exponents of the added functions. The active space is the same as that of the lower-level energy calculations, and the basis set employed here is the uncontracted aug-cc-pVQZ basis. The nonadiabatic terms, required for calculating for the RT coupling, are the matrix elements of electronic orbital angular momentum L, and they are obtained as expectation values over the SA-CASSCF wave functions.

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The RT coupling takes effect when the molecule approaches linearity, that is, lies on the z-axis, hence the matrix elements of Lx and Ly can be neglected. The details of the RT Hamiltonian have been discussed in another publication [52]. PES Grid. These were chosen carefully to accurately represent the dynamically important regions, particularly the vicinities of the minimum and near linearity.

As pointed out by Liu and co-workers [34], the 11n pair 31A' and 21A'' states conically cross not only the 11A pair 11A' and 11A'' states at around 3. We concentrate on the geometries with the CH bond length smaller than 3. Dozens of geometries are also selected for. In the important regions, points were computed with small increments of 0. In the calculations for the nonadiabatic coupling terms, almost the same grids were selected as those used in the higher-level calculations. The idea of the dual-level strategy is to use two levels of ab initio calculations so as to reduce the number of higherlevel points needed for fitting [43], and the basic scheme is as follows.

And it should be noted that the energies of the lower level are obtained from Vx0, instead of the ab initio calculations. Each of these steps will now be described in more details as follows.

### Potential‐energy surfaces of low‐lying states of HNO

The final adiabatic potential V is expressed as. For both electronic states, a Morse-type coordinate. Finally, cos 0 n - 9 was employed to describe the ZHCH bending:. The parameters ai, a2, 0, and the Qjk are determined by performing unequally weighted least squares fit to the ab initio data.

Analytical Representation of Vcore. The core correlation surface Vcore was constructed in a similar way to the construction of V0, and the surfaces can be expressed as. Analytical Representation of V0. For the analytical representation of V0, the calculated energy points were fitted to a three-body expansion in curvilinear coordinates, which. Analytical Representation of the Nonadiabatic Coupling Terms. To fit the matrix elements of Lz and LZ, into analytical representations, several types of polynomials have been tested, and the following one provides a good description of the nonadiabatic coupling terms, Lab, Laa, and Lbb:.

Several types of cos functions instead of 0 n - 9 were tested as angular coordinate, but very good results were not obtained. Many test calculations were performed with different polynomial orders M, and the dependences are shown in Table S1 see Table S1 in the Supplementary Material available online at doi: The complete set of parameters amounts to a total of linear coefficients and 3 nonlinear coefficients.

The fit for V0 has RMS errors of In the fit of Vcore, M is taken as 6. The complete set of parameters amounts to a total of 85 linear coefficients. The fit for Vcore has. RMS errors of The numerical values of all parameters to generate the surfaces and coupling terms reported in the present study are presented in Tables S2, S3, and S4. Adiabatic PESs. We found the fitted potentials to be smooth and without any artificial oscillations. The degeneracy of the two lowest-lying singlet adiabatic potentials is illustrated in Figure 2. The 21A'state, which lies between the 11A' and 31 A' state, is not shown.

The zero energy is taken at the CH2 11A' minimum. To further improve the behavior of our PESs at linear geometries, we used an assumed angle dependence switching function to smoothly connect the two states above at linearity. Thus, this function is defined as follows:. Two adiabatic potentials for the a1A1 and b1B1 states of CH2, which are going to be degenerate at linearity, are represented as follows:. From the experimental side, a wide number of studies have led to the determination of accurate equilibrium geometries for the a1A1 and b1B1 states of CH2.

Geometries and relative energies of minima obtained from our work along with the available experimental and other theoretical values are given in Table 2. Despite the ground singlet state being widely studied, there have been limited research on the first excited singlet state b1B1.

Table 2: Geometries and relative energies of the minima of the two lowest-lying singlet states of CH2. Re is the equilibrium interatomic distance, and 6e is the equilibrium ZHCH.

## Microscopic benchmark study of triaxiality in low-lying states of 76Kr - INSPIRE-HEP

As expected, its molecular orbitals and Mulliken atomic distributions are very similar to those of the ground state, which makes the ab initio calculation more difficult. The CV effects have a geometry dependence, and the shifts can be positive or negative. The core correlations varies from to cm The inclusion of core and core-valence correlation decreases the bond lengths by 0. RCH1 is fixed at 2.

While the energy changes with the CH bond stretched, the degeneracy of the 11A' and 11A'' or 31A' and 21A'' of CH2 is not lifted so long as the molecule is linear. The barrier to linearity plays a very important role in quantum mechanical calculations of vibronic energy levels when the RT effect is considered [18]. The height of the barrier to linearity in a1A1 CH2 has been a long standing source of controversy. The range of reported barrier heights for linearity in the a1A1 state of CH2 is quite large, varying from to cm-1, which is summarized in Table 3.

Herzberg and Johns originally gave a value of cm-1, estimated from the spacing of the b1B1 bending vibrational levels [2]. Duxbury and Jungen [56] obtained a barrier of. An earlier ab initio calculation gave a barrier height of cm-1 [57]. Kalemos et al. Green Jr. The value of cm-1 was obtained from the PESs constructed by Gu et al. Liu et al. Our ab initio calculation at the same level with five reference states gives the barrier to linearity as The core. It may be due to the fact that the Is electron is affected when the valence orbitals change from sp2 hybridization at the minimum area to sp hybridization at linearity.

Our PESs predict a height of Renner-Teller Nonadiabatic Coupling Terms. The polynomial order M is taken as 9. Figures 5 a , 5 b , and 5 c describe the stretching potential curves of the five singlet states 11A', 21A', 31A', 11 A'' and 21 A'' state of CH2 as functions of bond length Rch2, with Rch1 fixed at 2. Figure 7 shows the variation of the electronic matrix elements for the two lowest-lying singlet electronic states as functions of the bending angle ZHCH, with the CH-distance optimized for the a1A1 state. It is shown that, as the molecule bends, the curve ofLab decreases monotonically.

The energy levels are in cm-1, relative to the zero point energy of the a1A1 state. Vibronic Energy Level Calculations. We have calculated the vibronic energy levels of the a1 A1 and b1B1 states on our ab initio PESs employing the block improved relaxation scheme [61, 62] in the multiconfiguration time-dependent Hartree MCTDH method []. The variation of the electronic matrix elements with geometry is not considered, and other groups also adopted this treatment in previous calculations. For consistency, the energy levels in Table 4 are labeled by the bent molecular notation v1, v2, v3.

Our calculated results are in excellent agreement with the experimental values, reflecting the accuracy of the constructed ab initio PESs. For the energy level of S 0,2,0 , our result cm-1 is closer to the experimental value cm-1 from Sears et al. The experimental results from Sears et al. Compared with the. Free online A O math quiz we will practice various types of questions on math. But because the flow is non-isentropic, the total pressure downstream of the shock is always less than the total pressure upstream of the shock.

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