Review is based on critical analysis of the paper "Tomographic imaging of molecular orbitals" by J. Itatani et al.(More)
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Applying power of computers to solve chemical problems
Aharonov-Bohm effect is one of the most fascinating discoveries of the 20th century.
Here is a typical problem where Aharonov-Bohm effect appear:
An electron is inside of a sphere with radius 5 Å. Mass and thickness of the sphere are negligible. However, electromagnetic fields freely pass through the sphere without reflection or attenuation. The shell is confined to a circular track of radius 50 Å. A solenoid of radius 1 Å is perpendicular to the track, passing through its center. A magnetic filed inside of the solenoid is B and 0 outside. Describe the eigenstates of the system.
At first glance it seems simple. But how would you solve it?
The solution will be provided in the next post - stay tuned!
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Paper "Tomographic imaging of molecular
orbitals" by J. Itatani. in Nature challenges common QM (quantum
mechanics) interpretation of orbitals as pure mathematical constructs.
Authors goes as far as to say that they experimentally measured HOMO (highest occupied molecular orbital). Is that possible indeed or is it just a wrong interpretation of the experiment?
The paper could be found at:
I appreciate your comments on this issue.
You can vote on this question here:
[UPDATE] I've published critical review of this paper in another post:
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My article finally has been published in "Theoretical Chemistry Accounts"!
Title: Structure, vibrational frequencies, ionization energies, and photoelectron spectrum of the para-benzyne radical anion
Abstract: Equilibrium structure, vibrational frequencies, and ionization energies of the para-benzyne radical anion are characterized by coupled-cluster and equation-of-motion methods. Vibronic interactions with the low-lying excited state result in a flat potential energy surface along the coupling mode and even in a lower-symmetry C2v structures. Additional complications arise due to Hartree–Fock instabilities and near-instabilities. The magnitude of vibronic interactions was characterized by geometrical parameters, charge localization patterns and energy differences between the D2h and 2v structures. The observed trends suggest that the C2v minimum predicted by several theoretical methods is an artifact of incomplete correlation treatment. The comparison between the calculated and experimental spectrum confirmed D2h structure of the anion, as well as accuracy of the coupled-cluster and spin-flip structures, frequencies and normal modes of the anion and the diradical. Density functional calculations (B3LYP) yielded only a D2h minimum, however, the quality of the structure and vibrational frequencies is poor, as follows from the comparison to high-level wave function calculations and the calculated spectrum. The analysis of charge localization patterns and the performance of different functionals revealed that B3LYP underestimates the magnitude of vibronic interactions due to self-interaction error.
Keywords: Para-benzyne radical anion, Photoelectron spectroscopy, Coupled-cluster methods, Density functional theory, Symmetry breaking.
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List of quantum chemistry abbreviations is very useful for starting theoreticians and especially for experimentalists who need to read a quantum chemistry paper.
AO - atomic orbital
CAS - complete active space
CASSCF - complete active space self-consistent-field
CCSD - coupled-cluster theory with single and double excitations
CI - configuration interaction
CISD - configuration interaction with single and double excitations
CLSCF - closed-shell self-consistent-field
CSF - configuration stat function
DFT - density functional theory
DIIS - Direct Inversion in the Iterative Subspace
DM - Direct Minimization
EOM - equation of motion
EA - electron attachement
GDM - geometric direct minimization
GTO - Gaussian-type orbital
GVB - generalized valence bond
HF - Hartree-Fock
HOMO - highest occupied molecular orbital
IP - ionization potential
LCAO - linear combination of atomic orbitals
LUMO - lowest unoccupied molecular orbital
MO - molecular orbital
MOM - maximum overlap method
MCSCF - multiconfiguration self-consitent-field
MP2 - second-order Møller-Plesset perturbation theory
NBO - natural bond orbital
NBO - non-bonding orbital
NVT - nuclear vibrational theory
OO - optimized orbital
PES - potential energy surface
RHF - restricted Hartree-Fock
ROHF - restricted open shell Hartree-Fock
SCF - self-constistent-field
SF - spin flip
STO - Slater-type orbital
TDDFT - time–dependent density functional theory
TOSH - transition–optimized shifted Hermite
TZ - triple zeta
UHF - unrestricted Hartree-Fock
VCI - vibrational configuration interaction
VPT2 - second–order vibrational perturbation theory
ZPE - zero point energy
If you have any additions to this list please fill free to commnet on that.
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I have just finished configuration of our new computational cluster (11 powerful computers) for quantum chemistry calculations. Here is how it looks like:
Computational Cluster: Front View
Computational Cluster: Rear View
OS: Scientific Linux 4.4
Quantum chemistry packages: Gamess, Molpro, Q-Chem
Server (CPU: 2 x Xeon 5160 3GHz, RAM: 8GB, HDD: 4 x 300GB)
8 x Regular nodes (CPU: 2 x Xeon 5160 3GHz, RAM: 16GB, HDD: 4 x 300GB)
2 x Heavy nodes (CPU: 2 x Xeon 5160 3GHz, RAM: 24GB, HDD: 6 x 300GB)
The most interesting feature of this cluster is network boot - all nodes do not have OS installed, instead they boot from the server. This make configuration much more manageable.
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My recent studies of para-benzyne radical anion has shown that relative energy difference between C2v and D2h structures calculated at DFT level of theory are proportional to the electron self-interaction error included in DFT potential. The more HF exchange DFT potential has the smaller is the electron self-interaction error. Thus the most common B3LYP potential places D2h structure lower in energy mostly because of the electron self-interaction error.
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Q.I.) Why different authors assign different characters to the same orbital or normal vibrational mode? What is the source of this differences?
If different orientations of the molecule are used, symmetry labels corresponding to the same orbital or mode may be different.
For example for the water molecule which symmetry axis coincide with Z axis, we still have two choices for the molecular plane: XZ and YZ, presented on the picture below.
|Two different orientations of water molecule|
Following table shows symmetry labels of water normal vibrational modes for these two different orientations.
|Mode||Picture||Irrep for XZ||Irrep for YZ|
The difference in labeling for asymmetric stretch directly follows from the definition of irreps for the C2v point group given in the table below.
|A1||1||1||1||1||z||x2, y2, z2|
As we can see b1 and b2 irreps differ in the way they change sign on reflections in XZ and YZ planes. Thus different selections of molecule plane (XZ or YZ) lead to different labeling of asymmetric stretch (b2 or b1).
In order to solve these ambiguity problem the symmetry conventions were developed.
Q.II.) What are the conventions for symmetry notations in computational chemistry and spectroscopy?
Conventions of the symmetry notations are given in the paper:
R. S. Mulliken,. J. Chem. Phys., 23, (1955) 1997
The most important symmetry conventions are summarized below:
Lower case letters should be used to describe normal vibrational modes and orbitals. E.g. ag normal mode
Upper case letters should be used to describe vibrational, vibronic and electronic states. E.g. Ag electronic state
Character symbols and their definitions should be used exactly as given in G. Herzberg, "Infrared and Raman Spectra of Polyatomic Molecules (New York, 1945), besides T and t symbols should be used instead of F and f for triply-degenerate species.
For the following symmetry point groups molecules should be oriented as describe below:
1) planar C2v molecules: X-axis perpendicular to the plane of molecule, Z-axis is axis of symmetry
2) planar D2h molecules: X-axis perpendicular to the plane of molecule, Z-axis passes through the greatest number of atoms. If the last conditions is not unambiguous Z-axis should pass through the greatest number of bonds. For other symmetry point groups read the original paper.
Normal vibrational modes should be labeled following next rules:
1) Modes should be grouped in blocks with the same character.
2) Blocks with different characters should be ordered as in Herzberg's book (p. 272)
3) Modes inside of each block should be ordered from the highest to the lowest
Q.III.) What information should be presented to make symmetry assignment unambiguous?
The only way to make the symmetry assignment unambiguous for the reader is either to follow standard conventions (preferred) or to specify orientation of the molecule in space.
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