scholarly journals Manifold curvature and Ehrenfest forces with a moving basis

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Jessica Halliday ◽  
Emilio Artacho
Keyword(s):  
Extrinsic Curvature ◽  
Quantum States ◽  
Basis Set ◽  
Electronic Subsystem ◽  
Intrinsic Curvature ◽  
Geometrical Meaning

Known force terms arising in the Ehrenfest dynamics of quantum electrons and classical nuclei, due to a moving basis set for the former, can be understood in terms of the curvature of the manifold hosting the quantum states of the electronic subsystem. Namely, the velocity-dependent terms appearing in the Ehrenfest forces on the nuclei acquire a geometrical meaning in terms of the intrinsic curvature of the manifold, while Pulay terms relate to its extrinsic curvature.

scholarly journals Characterization of the pseudo-symmetries of ideal Wintgen submanifolds of dimension 3

2010 ◽  
Vol 88 (102) ◽  
pp. 53-65
Author(s):  
Ryszard Deszcz ◽  
Miroslava Petrovic-Torgasev ◽  
Zerrin Şentürk ◽  
Leopold Verstraelen
Keyword(s):  
Extrinsic Curvature ◽  
Real Space ◽  
Point Of View ◽  
Intrinsic Curvature ◽  
Space Forms ◽  
3 Dimensional ◽  
Arbitrary Codimension ◽  
Sectional Curvatures ◽  
Wintgen Ideal Submanifolds ◽  
Normalized Scalar Curvature

Recently, Choi and Lu proved that the Wintgen inequality ? ? H2??? +k, (where ? is the normalized scalar curvature and H2, respectively ??, are the squared mean curvature and the normalized scalar normal curvature) holds on any 3-dimensional submanifold M3 with arbitrary codimension m in any real space form ~M3+m(k) of curvature k. For a given Riemannian manifold M3, this inequality can be interpreted as follows: for all possible isometric immersions of M3 in space forms ~M3+m(k), the value of the intrinsic curvature ? of M puts a lower bound to all possible values of the extrinsic curvature H2 ? ?? + k that M in any case can not avoid to ?undergo" as a submanifold of ?M. From this point of view, M is called a Wintgen ideal submanifold of ~M when this extrinsic curvature H2 ??? +k actually assumes its theoretically smallest possible value, as given by its intrinsic curvature ?, at all points of M. We show that the pseudo-symmetry or, equivalently, the property to be quasi-Einstein of such 3-dimensional Wintgen ideal submanifolds M3 of M~3+m(k) can be characterized in terms of the intrinsic minimal values of the Ricci curvatures and of the Riemannian sectional curvatures of M and of the extrinsic notions of the umbilicity, the minimality and the pseudo-umbilicity of M in ~M.

scholarly journals Weingarten Type Surfaces in ℍ2 × ℝ and 𝕊2 × ℝ

2017 ◽  
Vol 69 (6) ◽  
pp. 1292-1311
Author(s):  
Abigail Folha ◽  
Carlos Peñafiel
Keyword(s):  
Extrinsic Curvature ◽  
Product Spaces ◽  
Intrinsic Curvature ◽  
Complete Surfaces

AbstractIn this article, we study complete surfaces Σ, isometrically immersed in the product spaces ℍ2 × ℝ or 𝕊2 × ℝ having positive extrinsic curvature Ke . Let Ki denote the intrinsic curvature of Σ. Assume that the equation aKi + bKe = c holds for some real constants a ≠ 0,b >0, and c. The main result of this article states that when such a surface is a topological sphere, it is rotational.

Geometry of Quantum States

2006 ◽  
Author(s):  
Ingemar Bengtsson ◽  
Karol Zyczkowski
Keyword(s):  
Quantum States

Transport of quantum states of periodically driven systems

1990 ◽  
Vol 51 (8) ◽  
pp. 709-722 ◽  
Author(s):  
H.P. Breuer ◽  
K. Dietz ◽  
M. Holthaus
Keyword(s):  
Quantum States ◽  
Driven Systems ◽  
Periodically Driven

Quantum States & Processes

1994 ◽  
Vol 187 (Part_1) ◽  
pp. 156-156
Author(s):  
H.-J. Unger
Keyword(s):  
Quantum States

Density functional theory, Comparative vibrational spectroscopic studies, HOMO-LUMO, and NBO analysis of Trichloro isocyanuric acid and Trithio cyanuric acid

2015 ◽  
Vol 8 (3) ◽  
pp. 2197-2221
Author(s):  
Theraviyum Chithambarathanu ◽  
M. Darathi ◽  
J. DaisyMagdaline ◽  
S. Gunasekaran
Keyword(s):  
Density Functional ◽  
Cyanuric Acid ◽  
Nbo Analysis ◽  
Basis Set ◽  
Potential Energy Distribution ◽  
Point Energy ◽  
Isocyanuric Acid ◽  
Ft Ir ◽  
Homo Lumo ◽  
Ft Raman

The molecular vibrations of Trichloro isocyanuric acid (C3Cl3N3O3) and Trithio cyanuric acid (C3H3N3S3) have been investigated in polycrystalline sample at room temperature by Fourier Transform Infrared (FT-IR) and FT-Raman spectroscopies in the region 4000-450 cm-1 and 4000-50 cm-1 respectively, which provide a wealth of structural information about the molecules. The spectra are interpreted with the aid of normal co-ordinate analysis following full structure optimization and force field calculations based on density functional theory   (DFT) using standard B3LYP / 6-311++ G (d, p) basis set for investigating the structural and spectroscopic properties. The vibrational frequencies are calculated and the scaled values are compared with experimental FT-IR and FT-Raman spectra. The scaled theoretical wave numbers shows very good agreement with experimental ones. The complete vibrational assignments are performed on the basis of potential energy distribution (PED) of vibrational modes, calculated with scaled quantum (SQM) method. Stability of the molecule arising from hyper conjugative interactions, charge delocalization has been analyzed using natural bond orbital (NBO) analysis. The results show that change in electron density (ED) in σ* and π* anti-bonding orbitals and second order delocalization   energy (E2) confirm the occurrence of Intra molecular Charge Transfer (ICT) within the molecule. The thermodynamic properties like heat capacity, entropy, enthalpy and zero point energy have been calculated for the molecule. The frontier molecular orbitals have been visualized and the HOMO-LUMO energy gap has been calculated. The Molecular Electrostatic Potential (MEP) analysis reveals the sites for electrophilic attack and nucleophilic reactions in the molecule.

Performance of Electronic Structure Methods for the Description of Hückel-Möbius Interconversions in Extended π-Systems

2019 ◽  
Author(s):  
Tatiana Woller ◽  
Ambar Banerjee ◽  
Nitai Sylvetsky ◽  
Xavier Deraet ◽  
Frank De Proft ◽  
...  
Keyword(s):  
Electronic Structure ◽  
Basis Set ◽  
Dft Methods ◽  
Molecular Switching ◽  
Wide Range ◽  
Electronic Structure Methods ◽  
Explicitly Correlated ◽  
Relative Errors ◽  
The Stability ◽  
Basis Set Expansion

<p>Expanded porphyrins provide a versatile route to molecular switching devices due to their ability to shift between several π-conjugation topologies encoding distinct properties. Taking into account its size and huge conformational flexibility, DFT remains the workhorse for modeling such extended macrocycles. Nevertheless, the stability of Hückel and Möbius conformers depends on a complex interplay of different factors, such as hydrogen bonding, p···p stacking, steric effects, ring strain and electron delocalization. As a consequence, the selection of an exchange-correlation functional for describing the energy profile of topological switches is very difficult. For these reasons, we have examined the performance of a variety of wavefunction methods and density functionals for describing the thermochemistry and kinetics of topology interconversions across a wide range of macrocycles. Especially for hexa- and heptaphyrins, the Möbius structures have a pronouncedly stronger degree of static correlation than the Hückel and figure-eight structures, and as a result the relative energies of singly-twisted structures are a challenging test for electronic structure methods. Comparison of limited orbital space full CI calculations with CCSD(T) calculations within the same active spaces shows that post-CCSD(T) correlation contributions to relative energies are very minor. At the same time, relative energies are weakly sensitive to further basis set expansion, as proven by the minor energy differences between MP2/cc-pVDZ and explicitly correlated MP2-F12/cc-pVDZ-F12 calculations. Hence, our CCSD(T) reference values are reasonably well-converged in both 1-particle and n-particle spaces. While conventional MP2 and MP3 yield very poor results, SCS-MP2 and particularly SOS-MP2 and SCS-MP3 agree to better than 1 kcal mol<sup>-1</sup> with the CCSD(T) relative energies. Regarding DFT methods, only M06-2X provides relative errors close to chemical accuracy with a RMSD of 1.2 kcal mol<sup>-1</sup>. While the original DSD-PBEP86 double hybrid performs fairly poorly for these extended p-systems, the errors drop down to 2 kcal mol<sup>-1</sup> for the revised revDSD-PBEP86-NL, again showing that same-spin MP2-like correlation has a detrimental impact on performance like the SOS-MP2 results. </p>

Divergence of Many-Body Perturbation Theory for Noncovalent Interactions of Large Molecules

2019 ◽  
Author(s):  
Brian Nguyen ◽  
Guo P Chen ◽  
Matthew M. Agee ◽  
Asbjörn M. Burow ◽  
Matthew Tang ◽  
...  
Keyword(s):  
Perturbation Theory ◽  
System Size ◽  
State Density ◽  
Second Order ◽  
Binding Energies ◽  
Basis Set ◽  
Many Body ◽  
Many Body Perturbation Theory ◽  
Relative Errors ◽  
Ground State Density

Prompted by recent reports of large errors in noncovalent interaction (NI) energies obtained from many-body perturbation theory (MBPT), we compare the performance of second-order Møller–Plesset MBPT (MP2), spin-scaled MP2, dispersion-corrected semilocal density functional approximations (DFA), and the post-Kohn–Sham random phase approximation (RPA) for predicting binding energies of supramolecular complexes contained in the S66, L7, and S30L benchmarks. All binding energies are extrapolated to the basis set limit, corrected for basis set superposition errors, and compared to reference results of the domain-based local pair-natural orbital coupled-cluster (DLPNO-CCSD(T)) or better quality. Our results confirm that MP2 severely overestimates binding energies of large complexes, producing relative errors of over 100% for several benchmark compounds. RPA relative errors consistently range between 5-10%, significantly less than reported previously using smaller basis sets, whereas spin-scaled MP2 methods show limitations similar to MP2, albeit less pronounced, and empirically dispersion-corrected DFAs perform almost as well as RPA. Regression analysis reveals a systematic increase of relative MP2 binding energy errors with the system size at a rate of approximately 1‰ per valence electron, whereas the RPA and dispersion-corrected DFA relative errors are virtually independent of the system size. These observations are corroborated by a comparison of computed rotational constants of organic molecules to gas-phase spectroscopy data contained in the ROT34 benchmark. To analyze these results, an asymptotic adiabatic connection symmetry-adapted perturbation theory (AC-SAPT) is developed which uses monomers at full coupling whose ground-state density is constrained to the ground-state density of the complex. Using the fluctuation–dissipation theorem, we obtain a nonperturbative “screened second-order” expression for the dispersion energy in terms of monomer quantities which is exact for non-overlapping subsystems and free of induction terms; a first-order RPA-like approximation to the Hartree, exchange, and correlation kernel recovers the macroscopic Lifshitz limit. The AC-SAPT expansion of the interaction energy is obtained from Taylor expansion of the coupling strength integrand. Explicit expressions for the convergence radius of the AC-SAPT series are derived within RPA and MBPT and numerically evaluated. Whereas the AC-SAPT expansion is always convergent for nondegenerate monomers when RPA is used, it is found to spuriously diverge for second-order MBPT, except for the smallest and least polarizable monomers. The divergence of the AC-SAPT series within MBPT is numerically confirmed within RPA; prior numerical results on the convergence of the SAPT expansion for MBPT methods are revisited and support this conclusion once sufficiently high orders are included. The cause of the failure of MBPT methods for NIs of large systems is missing or incomplete “electrodynamic” screening of the Coulomb interaction due to induced particle–hole pairs between electrons in different monomers, leaving the effective interaction too strong for AC-SAPT to converge. Hence, MBPT cannot be considered reliable for quantitative predictions of NIs, even in moderately polarizable molecules with a few tens of atoms. The failure to accurately account for electrodynamic polarization makes MBPT qualitatively unsuitable for applications such as NIs of nanostructures, macromolecules, and soft materials; more robust non-perturbative approaches such as RPA or coupled cluster methods should be used instead whenever possible.<br>

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