PT-Symmetry in Hartree–Fock Theory

<div> <div> <p>P T -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 P T 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 symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are P T -symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break PT -symmetry.</p> </div> </div>

PT-Symmetry in Hartree–Fock Theory

<div> <div> <p>P T -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 P T 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 symmetry-broken unrestricted HF wave functions (i.e. diradical configurations) are P T -symmetric, while the symmetry-broken restricted HF wave functions (i.e. ionic configurations) break PT -symmetry.</p> </div> </div>

A Study of the electronic structure of CdS Nanocrystals using density functional theory

Density Functional Theory at the generalized-gradient approximation level coupled with large unit cell method is used to simulate the electronic structure of (II-VI) zinc-blende cadmium sulfide nanocrystals that have dimensions 2-2.5 nm. The calculated properties include lattice constant, conduction and valence bands width, energy of the highest occupied orbital, energy of the lowest unoccupied orbital, energy gap, density of states etc. Results show that lattice constant and energy gap converge to definite values. However, highest occupied orbital, lowest unoccupied orbital fluctuates indefinitely depending on the shape of the nanocrystal.

Sodium Interaction with Disodium Terephthalate Molecule: an Ab Initio Study

ABSTRACTDisodium terephthalate (Na2TP), which is a disodium salt of terephthalic acid, is very promising organic electrode material for Na-ion batteries. We present an ab initio study of Na binding mechanism with Na2TP molecule. Specially, we provide the interaction energy of Na atom(s), effect of Na concentration on interaction energy, electronic properties of clean and Na attached Na2TP, and Na binding mechanism with Na2TP. We show that up to eight Na atoms can be attached to a single Na2TP molecule. The interaction energy of Na atoms varies from -0.79 to -0.66 eV with attachment of one to eight Na atoms. The adsorbed Na atom interacts with O atoms of carboxylate group and Na atoms of the salt molecule. The interaction between adsorbed Na and C atoms of the molecule is found to be not important for Na bindings. Attachment of a single Na atom generates a singly occupied orbital which becomes doubly occupied with attachment of second Na atoms. Attachment of more than two Na atoms leads to electron occupation of bonding orbitals formed between Na atoms and the carboxylate groups.

Synthesis, Photo-sensitive and Electrochemical Properties of Rod-like Aromatic Aldehyde with Azo Bridge

Eight new rod-like liquid crystal molecules composed by a long rigid core of three six-member rings (cyclohexane ring or benzene ring), azo, ester and terminal aldehyde groups have been prepared. These rod-like liquid crystalline molecules were designed to construct new structures to further study photo-isomerization in their mesophases. All the compounds have been characterized based on their basic spectral data, differential scanning calorimeter (DSC) and hot stage polarizing optical microscope (HS-POM). The result showed that all the molecules, even those with the shortest terminal methyl group, have liquid crystalline properties. Their mesophases are nematic within the temperature ranges from 85 to 145°C. They exhibit photo-sensitivities not only in methanol solutions but also in a mesophase when exposed to UV light. The highest occupied orbital (HOMO) and the lowest unoccupied orbital (LUMO) and the differences between the frontier molecular orbitals (E

Hyperconjugation in diethyl ether cation versus diethyl sulfide cation

Electron donation from the CH bond to the single occupied orbital is observed through the large red shift in the CH stretching band for the diethyl ether cation.

Theoretical study of phenol and hydroxyl radical reaction mechanism in aqueous medium by the DFT/B3LYP/6-31+G(d,p)/CPCM model

Four different possible reaction pathways of phenol and hydroxyl radical reaction were investigated theoretically by density functional theory (DFT) B3LYP with the 6-31+G(d,p) calculations under the conductor-like polarized continuum model (CPCM). According to frontier molecule orbital theory, both the highest occupied orbital and lowest occupied orbital of phenol (25th orbital) showed –602.79 and –43.53 kJ mol−1 molecular orbital energies, respectively. This resulted in a 559.27 kJ mol−1 relative energy gap. Relative energies of the ortho product radical (o-PR) (i.e., –54.08 kJ mol−1) was lower than those of both the para product radical (p-PR) (–50.03 kJ mol−1) and the meta product radical (m-PR) (–47.10 kJ mol−1). Then, o-PR was found to be the energetically most stable product radical. The ortho addition reaction path was confirmed as the most possible reaction path and its major intermediate was found as catechol with 99.09% product distribution. Percentages of hydroquinone, resorcinol, and phynoxyl radicals in the system were found as 0.053%, 0.029%, and 0.009%, respectively.