Theoretical analysis of electronic delocalization

The Block-Localized Wavefunction (BLW) Method and Its Applications

The reduced local Hartree–Fock energy for the neutral atoms Li through to Al have been calculated using the Hartree–Fock wave functions of Clementi and Roetti. These wave functions were found to give rather accurate values of the reduced local energy in the medium range region (r values 1∼12 a.u.9), but inaccurate values close to the nucleus. The reduced local energy was found in each case to exhibit oscillations about the ’’exact’’ total Hartree–Fock energy as a function of the configuration space coordinate r, indicating some cancellation of opposing errors in the wave functions used for the calculation of the total energy. A method is outlined for determining Hartree–Fock wave functions which not only satisfy the customary condition that the total energy be minimized, but in addition, are locally accurate. The criterion which is used to monitor the local accuracy is the reduced local energy. The reduced local energy is incorporated into a global accuracy constraint on the wave function, and the energy is determined subject to the minimization of this additional constraint. Calculations are performed for various helium wave functions. The five-term Clementi–Roetti wave function is refined, maintaining essentially the same total energy, but with considerable improvement in local accuracy. The Clementi (1965) wave function for helium is also refined. A ten-term function for helium is devised which has improved accuracy at the local level. Comparison of expectation values of r1k using the different wave functions is made, in order to assess the utility of the reduced local energy as a criterion for refinement of such values.

Recent papers on electron-N/sub 2/ scattering by Rumble, Stevens, and Truhlar (J. Phys. B 17, 3151 (1984)) and Weatherford, Brown, and Temkin (Phys. Rev. A 35, 4561 (1987)) have suggested that Hartree-Fock (HF) wave functions may not be accurate for calculating potentials for use in studying electron-molecule collisions. A comparison of results for electron-H/sub 2/ scattering using both correlated and HF wave functions is presented. It is found that for both elastic and inelastic collisions and for all energies considered (up to 10 eV) the HF wave functions yield results in excellent agreement with those obtained from the more accurate wave functions.

The half‐projected Hartree–Fock wave function (HPHF) is one of simplest models for introducing some electronic correlation effects. In this model, the wave function is built up with only two Slater determinants. This simple form suggests its application for the direct determination of singlet excited states. On the other hand, because the HPHF model does not mix singlet and triplet states with Ms = 0, it can be used for determining independently singlet and triplet states without any mutual contamination. In the present work, we applied this model to determine nine electronic states of the lithium molecule; one of them exhibits even the same symmetry of the fundamental one. For this purpose, the 6‐311G(d) basis was used. Potential energy curves were determined and some spectroscopic constants derived. The numerical results were compared with the available experimental data, as well as with other theoretical values. © 1995 John Wiley & Sons, Inc.

We report fixed-node diffusion Monte Carlo (FN-DMC) calculations of the singlet n-->pi( *) (CO) vertical transition of acrolein. The impact of the fixed-node approximation on the excitation energy is investigated. To do that, trial wave functions corresponding to various nodal patterns are used. They are constructed by using either a minimal complete-active-space self-consistent field (CASSCF) calculation involving an oxygen lone pair n and the pi( *) (CO) molecular orbitals or a more complete set involving all the molecular orbitals expected to play a significant role in the excitation process. Calculations of both states have been performed with molecular orbitals optimized separately for each state via standard "state specific" CASSCF calculations or by using a common set of optimized orbitals ["state averaged" CASSCF calculations] whose effect is to introduce some important correlation between the nodal patterns of the two electronic states. To investigate the role of the basis set three different basis of increasing size have been employed. The comparative study based on the use of all possible combinations of basis sets, active spaces, and type of optimized molecular orbitals shows that the nodal error on the difference of energies is small when chemically relevant active space and state-averaged-type CASSCF wave functions are used, although the fixed-node error on the individual total energies involved can vary substantially. This remarkable result obtained for the acrolein suggests that FN-DMC calculations based on a simple strategy (use of standard ab initio wave functions and no Monte Carlo optimization of molecular orbital parameters) could be a working computational tool for computing electronic transition energies for more general systems.

The aim of this work is to evaluate the one- electron expectation value from the radial electronic density function D(r1) for different wave function for the 2S state of Be atom . The wave function used were published in 1960,1974and 1993, respectavily. Using Hartree-Fock wave function as a Slater determinant has used the partitioning technique for the analysis open shell system of Be (1s22s2) state, the analyze Be atom for six-pairs electronic wave function , tow of these are for intra-shells (K,L) and the rest for inter-shells(KL) . The results are obtained numerically by using computer programs (Mathcad).

The aim of this work is to evaluate the one- electron expectation value from the radial electronic density function D(r1) for different wave function for the 2S state of Be atom . The wave function used were published in 1960,1974and 1993, respectavily. Using Hartree-Fock wave function as a Slater determinant has used the partitioning technique for the analysis open shell system of Be (1s22s2) state, the analyze Be atom for six-pairs electronic wave function , tow of these are for intra-shells (K,L) and the rest for inter-shells(KL) . The results are obtained numerically by using computer programs (Mathcad).

The unrestricted Hartree-Fock (UHF) wave functions for H- and for Li- have the Z+1th electron at infinity, and thus have the same energy as the neutral atoms. That is, the stability of these negative ions is not accounted for by Slater determinant wave functions, not even if the orbitals are allowed to split. We show that the difficulty here is that the UHF wave functions do not have the proper spin symmetry. If the Slater determinant is spin-projected and the orbitals optimized after projection (to obtain what is called the GF wave function), these negative ions are predicted correctly to be stable. Since the GF wave function leads to an independent particle interpretation, we see that instantaneous polarization of the neutralatom orbitals by the Z+1th electron is not crucial to the stability of these negative ions. From an analysis of the differences between the UHF and GF wave functions, we find that the key term leading to stability of the negative ions is an exchange term (particularly the nuclear attraction part of this term), just like the exchange term important in the valence-bond wave function of H2.

Ionic liquids (ILs) hold great promise in many fields including energy storage and generation, mechanical, pharmaceutical, synthetic and separation applications to name just a few. For any given application, the desired physical properties of the ideal IL may differ substantially from others and no widely applicable patterns or trends to facilitate intuitive design. The origins of physical properties lie in the characteristics of the intermolecular energetics, which consist of a complex interplay between electrostatic and dispersion forces. This thesis investigates and develops computational methodologies for calculating a reliable description of the intermolecular interactions for this challenging class of solvents and electrolytes of the future. The electrostatic approximation used in classical molecular dynamics (MD) where atomic partial charges are assigned was investigated in terms of methods based on density matrix partitioning, and the restrained electrostatic potentials (RESP) approaches. It was found that the “geodesic” atomic partial charge scheme, part of the RESP family, produced the most accurate charges. This was measured in terms of (a) charge convergence with increasing basis set size; (b) charge invariance with changes to the coordinate system; (c) insensitivity to minor structural changes on the resulting charges; (d) adequate capture of charge transfer effects; and (e) the preservation of symmetric of charges in symmetric molecules. Although charges can vary dramatically depending on the scheme used, the careful use of atomic partial charge schemes may still produce reliable forcefields, or at least serve as a rapid diagnostic tool to quantify electrostatic interactions and charge transfer. In moving towards unbiased a priori descriptions of IL intermolecular interactions, second-order Moller-Plesset perturbation theory (MP2) was used with the linear-scaling fragment molecular orbital (FMO) framework to assess the extent to which dispersion forces play a role in the intermolecular energetics. ILs of increasing size were examined such that the many-body effects may be captured. The dispersion energy contribution formed up to 20% of the total interaction energy. Furthermore, the interaction energy as produced by FMO was within 1 kJ mol−1 of the full-wavefunction MP2 interaction energy when three-body effects were included. As the dispersion interaction is purely a quantum mechanical phenomenon, correlated quantum mechanical methods, such as MP2 or coupled-cluster approaches, are required to provide an unbiased account of these effects. Ab initio methods such as MP2 and CCSD(T) scale formally as N⁵ an N⁷, respectively, with chemical system size. While the FMO approach provides a marked improvement in efficiency, the counterpoise (CP) approach to correcting the basis set superposition error (BSSE) is not amenable to fragmented approaches and requires each ion in the cluster to be calculated in the basis set of the entire system. In order to remove this bottleneck, the spin-component scaled second-order Moller Plesset perturbation theory (SCS-MP2) methodology was refined by fitting 174 non-CP corrected interaction energies at the MP2/cc-pVTZ level of theory to CP corrected CCSD(T)/CBS benchmark energies. This has resulted in an implicit BSSE correction that may be used within the highly efficient FMO framework, and is shown to yield results on par with or exceeding the accuracy of MP2/cc-pVQZ for clusters of two and four ion pairs (IPs). This new approach as been termed SCS-IL-MP2. An alternative dispersion corrected density functional theory (DFT) approach, DFT- D3, was assessed and refined in view of producing accurate interaction energies at the same CCSD(T)/CBS quality. The same test set of 174 ILs was used to fit the SCS-IL-MP2 approach was used to refit the DFT-D3 approach for both the Hartree-Fock (HF) wavefunction and the PBE and BLYP density functionals (DFs). In most cases, the selection of the DF and associated DFT-D version 3 (DFT-D3) parameters differed negligibly with all reaching within 1 to 2 kJ mol−1 per IP. HF-D3 parameters, on the other hand, showed a substantial improvement, particularly when used with the Becke-Johnson (BJ) damping function. Refitted HF-D3 and the BJ damping function was able to consistently provide interaction energy errors below 5 kJ mol−1 per IP. It would be worthwhile further investigating the application of the refined HF-D3 in its ability to produce reliable energies and geometries over a more diverse set of ILs. Both the SCS-IL-MP2 and new DFT-D3 approaches may be applied to the highly scal- able FMO framework. These form the core elements of the quantum chemistry toolbox for the study and understanding of the physicochemical properties of ILs that have so far been only superficially characterised by electronic structure theory. From this starting point, a rigorous and unbiased a priori understanding of the intermolecular interactions and resulting physicochemical properties may be predicted by means of efficient ab initio molecular dynamics (AIMD) techniques.

NCCL– anions (L = N2, CO and CS) exhibitnotable geometric differences: NCCNN– favors a bentstructure, whereas NCCCO– and NCCCS– prefer linear configurations. Here, we investigate the origin oftheir geometric differences using the conventional density functionaltheory (DFT) and block-localized wave function (BLW) method. Computationsreveal that the bent structure is preferred for all species at theBLW state in which the in-plane π// back-donationfrom NCC– to ligands is disabled. For NCCNN–, enforcing linearity leads to a gain in the π// electron delocalization (conjugation) stability, but itis insufficient to offset the steric penalty. Differently, NCCCO– and NCCCS– experience reduced stericrepulsion in linear geometries, thereby favoring more linear geometriesas their energy minima. “In situ” orbitalcorrelation diagrams reveal an orbital swap in NCC– with the approaching of the ligand L and confirm the carbone theoryproposed by the Frenking group. Overall, the geometries (or the degreeof bending) of NCCL– anions are governed by thebalance between the π// conjugation and steric repulsion.Substituting the electron-withdrawing NC group with CH3 or F, which provides in-plane σ-electrons or lone pairs, furtherbends the ∠RCL angle in RCNN– anions, whileRCCO– and RCCS– retain their nearlylinear forms.

The Huckel’s rule, Baird’s rule, and electronic shell closure model are classical and well-established concepts in chemistry, which have long been employed in rationalizing the aromaticity/antiarom...

Using the projected Hartree-Fock (HF) wave functions, a theoretical formulation is given for calculating the electromagnetic properties of nuclei, the nuclear form factors in electron scattering, $\mathrm{ft}$ values in $\ensuremath{\beta}$ decay, and the reduced widths of nuclear levels in direct reactions. The formulation is applied to calculate the quadrupole and magnetic moments, the electromagnetic transition probabilities, and the lifetimes of nuclear levels of a large number of nuclei ranging from $^{19}\mathrm{F}$ to $^{37}\mathrm{Cl}$ in the $2s\ensuremath{-}1d$ shell. The good agreement between the calculated and the experimental values indicates a remarkable success of the projected HF wave functions in predicting the electromagnetic properties of nuclei.

Hartree—Fock wavefunctions are presented for the LiH(X1Σ+), BeH(X2Σ+), BH(X1Σ+), CH(X2Πr), NH(X3Σ−), OH(X2Πi), and HF(X1Σ+) molecules. These are the analytic self-consistent-field wavefunctions obtained from the solutions of the Hartree—Fock—Roothaan equations. Large sets of Slater-type functions centered on both nuclei were used as the expansion basis, and extensive optimization of the orbital exponents has been carried out. The total energies obtained for Re(exptl) are −7.98731, −15.15312, −25.13137, −38.27935, −54.97806, −75.42083, and −100.07030 hartrees, respectively, for the AH hydrides listed above. The first ionization potentials, which are obtained from the Hartree—Fock energy differences between AH and AH+ systems, are 7.02, 8.14, 8.45, 10.08, 12.82, 11.44, and 14.54 eV, respectively. In addition, potential curves, spectroscopic constants, and certain other energetic quantities are presented. Crude estimates of the correlation energy of the first-row hydrides are made and such quantities are compared within the series and with their respective united and separated atoms. These results suggest that the changes in correlation energy of AH relative to the correct united atom is independent of which hydride is involved and the change is small.

The solvation of trans amides was investigated by the use of full gradient optimization ab initio quantum mechanical calculations techniques (HONDO 7) [1]. The complexes have been determined at the Hartree Fock (HF) level with 4–31 G* basis set and at the second order Moller-Plesset perturbation (MP2) level. Three NMA-water clusters were investigated (Figure 1): trans-NMA with one molecule of water at the CO group and one at the NH group (complex (A)) [2]; trans-NMA with two molecules of water forming a ring cluster at the amide oxygen (complex (B)); trans-NMA with two molecules of water at the amide oxygen forming hydrogen bonds along the direction of the lone-pair electrons (complex (C)) [3]. The \(C = O \cdots H(W)\) hydrogen bond lengths (Table 1), electron density population analysis and molecular orbital analysis demonstrate the importance of cooperative hydrogen bond interactions of the bound molecules of H2O. Calculations at the HF level indicate that complex (B) with a nonplanar amide bond and a chiral center at the amide nitrogen is more stable than complex (C). The stability of complexes is reversed at the MP2 level. Orbital analysis suggests that there is considerable solute occupied space reorganization caused by the rearrangement of the water solvent molecules.

PT-symmetry-invariance with respect to combined space reflection P and time reversal T-provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. PT-symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for PT-symmetry in the context of electronic structure theory and, specifically, within the Hartree-Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the PT operator if and only if the effective Fock Hamiltonian is PT-symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under PT, then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly PT-symmetric Slater determinants by forming PT-doublets (i.e., pairing each occupied orbital with its PT-transformed analogue), allowing PT-symmetry to be conserved throughout the self-consistent process. Finally, considering the H2 molecule as an illustrative example, we observe PT-symmetry in the HF energy landscape and find that the spatially symmetry-broken unrestricted HF wave functions (i.e., diradical configurations) are PT-symmetric, while the spatially symmetry-broken restricted HF wave functions (i.e., ionic configurations) break PT-symmetry.