Matrices and Wave Functions under Double-Group Symmetry

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Time Reversal ◽  
Point Group ◽  
Computational Effort ◽  
Wave Functions ◽  
Group Symmetry ◽  
Time Reversal Symmetry ◽  
Point Group Symmetry ◽  
Matrix Elements ◽  
Double Group

Symmetry is one of the most versatile theoretical tools of physics and chemistry. It provides qualitative insight into the wave functions and properties of systems, and it has also been used successfully to obtain great savings in computational efforts. In the preceding chapter we examined time-reversal symmetry, and now we turn to the more familiar point-group symmetry. We show how relativity requires special consideration and extensions of the concepts developed for the nonrelativistic case, and how time-reversal symmetry and double-group symmetry are connected. Although the techniques that incorporate double-group symmetry presented here are primarily aimed at four-component calculations, they are equally applicable to two-component calculations in which the spin-dependent operators are included at the SCF stage of a calculation. In the preceding chapter, we have shown how the use of time-reversal symmetry can lead to considerable reduction in the number of unique matrix elements that appear in the operator expressions. However, we are also interested in the overall structure of the matrices of the operators. In particular, we are interested in possible block structures, where classes of matrix elements may be set to zero a priori. If the matrices can be cast in block diagonal form, we may save on storage as well as computational effort in solving eigenvalue problems, for example. Matrix blocking will already be effected by the point-group symmetry of the molecule.

Operators, Matrix Elements, and Wave Functions under Time-Reversal Symmetry

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Quantum Chemistry ◽  
Time Reversal ◽  
Relativistic Quantum ◽  
Spin Symmetry ◽  
Computational Effort ◽  
Wave Functions ◽  
Time Reversal Symmetry ◽  
Molecular Systems ◽  
Self Consistent Field ◽  
Relativistic Quantum Chemistry

We now take on the task of developing the theory and methods for a relativistic quantum chemistry. The aim is to arrive at a qualitative as well as a quantitative understanding of the relativistic effects in molecules. We must be able to predict the effects of relativity on the wave functions and electron densities of molecules, and on the molecular properties arising from these. And we must develop methods and algorithms that enable us to calculate the properties and interactions of molecules with an accuracy comparable to that achieved for lighter systems in a nonrelativistic framework. Parts of this development follow fairly straightforwardly from our considerations of the atomic case in part II, but molecular systems represent challenges of their own. This is particularly true for the computational techniques. From the nonrelativistic experience we know that present-day quantum chemistry owes much of its success to the enormous effort that has gone into developing efficient methods and algorithms. This effort has yielded powerful tools, such as the use of basis-set expansions of wave functions, the exploitation of molecular symmetry, the description of correlation effects by calculations beyond the mean-field approximation, and so on. In developing a relativistic quantum chemistry, we must be able to reformulate these techniques in the new framework, or replace them by more suitable and efficient methods. In nonrelativistic theory, spin symmetry provides one of the biggest reductions in computational effort, such as in the powerful and elegant Graphical Unitary Group Approach (GUGA) for configuration interaction (CI) calculations (Shavitt 1988). For relativistic applications, time-reversal symmetry takes the place of spin symmetry, and this chapter is devoted to developing a formalism for efficient incorporation of this symmetry in our theory and methods. Time-reversal symmetry includes the spin symmetry of nonrelativistic systems, but there are significant differences from spin symmetry for systems with a Hamiltonian that is spin-dependent. The development of techniques that incorporate time-reversal symmetry presented here are primarily aimed at four-component calculations, but they are equally applicable to two-component calculations in which the spin-dependent operators are included at the self-consistent field (SCF) stage of a calculation.

Gas Phase Spin Relaxation of ZX3Y Molecules in the Dipolar Approximation

1975 ◽  
Vol 53 (7) ◽  
pp. 723-738 ◽  
Author(s):  
B. C. Sanctuary ◽  
R. F. Snider
Keyword(s):  
Kinetic Theory ◽  
Gas Phase ◽  
Magnetic Relaxation ◽  
Dipolar Coupling ◽  
Point Group ◽  
Group Symmetry ◽  
Subsequent Paper ◽  
Point Group Symmetry ◽  
Proper Treatment ◽  
Gas Kinetic Theory

The gas kinetic theory of nuclear magnetic relaxation of a polyatomic gas, as formulated in the previous paper, is evaluated for ZX3Y molecules relaxing via a dipolar coupling Hamiltonian. Stress is given to a proper treatment of point group symmetry, here C3v, and the possibility of molecular inversion is included. The detailed formula for the spin traces is however restricted to X nuclei with spin 1/2. A subsequent paper uses these results to elucidate the structure of the high density dependence of T1 forCF3H.

scholarly journals Spin-reversal energy barriers of 305 K for Fe2+ d6 ions with linear ligand coordination

Nanoscale ◽  
2017 ◽  
Vol 9 (30) ◽  
pp. 10596-10600 ◽  
Author(s):  
Lei Xu ◽  
Ziba Zangeneh ◽  
Ravi Yadav ◽  
Stanislav Avdoshenko ◽  
Jeroen van den Brink ◽  
...  
Keyword(s):  
Solid State ◽  
Magnetic Anisotropy ◽  
Point Group ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
State Matrix ◽  
Magnetic Anisotropy Energy ◽  
Spin Reversal ◽  
Ligand Coordination ◽  
Li Ion

A remarkably large magnetic anisotropy energy of 305 K is computed by quantum chemistry methods for divalent Fe2+ d6 substitutes at Li-ion sites with D6h point-group symmetry within the solid-state matrix of Li3N.

Tunable double Weyl phonons driven by chiral point group symmetry

2021 ◽  
Vol 103 (10) ◽  
Author(s):  
Y. J. Jin ◽  
Z. J. Chen ◽  
X. L. Xiao ◽  
H. Xu
Keyword(s):  
Point Group ◽  
Group Symmetry ◽  
Point Group Symmetry

scholarly journals Dodecaallylhexasilacyclohexane

IUCrData ◽  
2017 ◽  
Vol 2 (6) ◽  
Author(s):  
Yoshiyuki Mizuhata ◽  
Yamato Omatsu ◽  
Norihiro Tokitoh
Keyword(s):  
Title Compound ◽  
Point Group ◽  
Chair Conformation ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
Compound C

The molecule of the title compound, C36H60Si6, exhibits point group symmetryCi, with the centre of inversion located at the centre of the Si6ring. The Si6ring has a chair conformation. In the crystal, molecules are linkedviaC—H...π(allyl) interactions.

scholarly journals Synthesis, structure determination and characterization by UV–Vis and IR spectroscopy of bis(diisopropylammonium) cis-dichloridobis(oxalato-κ2 O 1,O 2)stannate(IV)

2019 ◽  
Vol 75 (6) ◽  
pp. 742-745
Author(s):  
Bougar Sarr ◽  
Abdou Mbaye ◽  
Cheikh Abdoul Khadir Diop ◽  
Mamadou Sidibe ◽  
Yoann Rousselin
Keyword(s):  
Crystal Structure ◽  
Metal Ion ◽  
Point Group ◽  
Chain Structure ◽  
Chloride Ions ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
Chloride Dihydrate ◽  
Distorted Octahedral Coordination ◽  
Distorted Octahedral

The organic–inorganic title salt, (C6H16N)2[Sn(C2O4)2Cl2] or ( i Pr2NH2)2[Sn(C2O4)2Cl2], was obtained by reacting bis(diisopropylammonium) oxalate with tin(IV) chloride dihydrate in methanol. The SnIV atom is coordinated by two chelating oxalate ligands and two chloride ions in cis positions, giving rise to an [Sn(C2O4)2Cl2]2− anion (point group symmetry 2), with the SnIV atom in a slightly distorted octahedral coordination. The cohesion of the crystal structure is ensured by the formation of N—H...O hydrogen bonding between (iPr2NH2)+ cations and [SnCl2(C2O4)2]2− anions. This gives rise to an infinite chain structure extending parallel to [101]. The main inter-chain interactions are van der Waals forces. The electronic spectrum of the title compound displays only one high intensity band in the UV region assignable to ligand–metal ion charge-transfer (LMCT) transitions. An IR spectrum was also recorded and is discussed.

Pseudo-symmetry in multiple twinned crystals having M3M point group symmetry

1971 ◽  
Vol 2 (12) ◽  
pp. 3485-3486 ◽  
Author(s):  
Santiago Harriague ◽  
Harry A. Leibovich
Keyword(s):  
Point Group ◽  
Group Symmetry ◽  
Point Group Symmetry
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