Review is based on critical analysis of the paper "Tomographic imaging of molecular orbitals" by J. Itatani et al.

 (More)

Solution of the previously posted problem on Aharonov-Bohm effect. (More)

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!

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:
http://spectroscopy.mps.ohio-state.edu/institute/2007/corkum/nature_Tomography.pdf

I appreciate your comments on this issue.

You can vote on this question here:
http://www.webqc.org/chemicalforum/viewtopic.php?t=466

[UPDATE] I've published critical review of this paper in another post:
http://vitalii.chemicalblogs.com/2_computational_chemistry/archive/480_review_are_orbitals_real.html

I've recently returned from Hawaii. There I've made several shots of the ground colored with Fe2O3. Looks amazing ...

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 C_2v 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 D_2h and _2v structures. The observed trends suggest that the C_2v minimum predicted by several theoretical methods is an artifact of incomplete correlation treatment. The comparison between the calculated and experimental spectrum confirmed D_2h 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 D_2h 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.

Full text of the original pulication is available at http://www.springer.com

Full text of the final draft "Structure, vibrational frequencies, ionization energies, and photoelectron spectrum of the para-benzyne radical anion"

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.

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

Computers:

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.

Poster for the Western Spectroscopy conference (click to enlarge):

My recent studies of para-benzyne radical anion has shown that relative energy difference between C_2v and D_2h 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 D_2h structure lower in energy mostly because of the electron self-interaction error.

I've just updated my article regarding symmetry conventions.

Check this out: conventions for symmetry notations

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
Asymmetric stretch		b1	b2
Symmetric stretch		a1	a1
Bending		a1	a1

The difference in labeling for asymmetric stretch directly follows from the definition of irreps for the C_2v point group given in the table below.

Character table for the C_2v point group

	E	C2 (z)	σv(xz)	σv(yz)	linear, rotations	quadratic
A1	1	1	1	1	z	x2, y2, z2
A2	1	1	-1	-1	Rz	xy
B1	1	-1	1	-1	x, Ry	xz
B2	1	-1	-1	1	y, Rx	yz

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. a_g normal mode
  

  Upper case letters should be used to describe vibrational, vibronic and electronic states. E.g. A_g 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 C_2v molecules: X-axis perpendicular to the plane of molecule, Z-axis is axis of symmetry
  2) planar D_2h 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.

point group symmetry tables

Vibronic interactions and symmetry breaking in the 1,4-benzyne radical anion

Frontier orbitals of 1,4-benzyne radical anion. Look nice ...