Messages

1

General Questions and Answers / Wavelengths of eigenstates in Brillouin Zone

« on: March 18, 2011, 00:22 »

Fellows,
      I am now a doctorate in inorganic/physical chemistry and am working on a presentation that will be done at the American Physics Society Meeting, in Dallas, about the eigenstates of graphene.  Although my presentation is in good shape right now, I have a question regarding a basic tenet of information, the wavelength of eigenstates that relate to crystal momentum.

For example, at Gamma, the wavelength is taken to be infinite, as there is no momentum to speak of.  I use Gamma eigenstates of regular and larger unit cells to describe the eigenstates in graphene for different k-points, transferring the wavelengths of the original eigenstates onto the Gamma eigenstates of the computed unit cells (1x1, 2x2, or 3x3).  This simplifies description of eigenstates significantly, and is the central point of my presentation.  The wavelength is used to describe phase relationships, and in Gamma eigenstates, there is no phase change anywhere other than individual orbitals inside the unit cell.  Going across crystal boundaries does not produce a phase change in all of the orbitals present inside the unit cell.  This makes the wavelength infinite in Gamma eigenstates, as the unit cell contents are used to represent the phase relationships for wavelength in crystal momentum.  In other k-points, such as M and K, however, this changes considerably, as the orbitals acquire a finite wavelength as the crystal boundary is crossed over.  This is what produces the crystal momentum in M and K, the phase relationship is modulated across crystal boundaries.

Using larger unit cells causes zone-folding in energy diagrams, but is quite useful for creating wavefunctions without momentum, so you obtain the phase relationships as if they were Gamma eigenstates!

How do we use the expression for crystal momentum to describe wavelength of eigenstates, given that we know the coordinates for each k-point?  I would like to do this so I can correct some minor details involving wavelengths of eigenstates.  I need the wavelengths that the k-points, M and K, represent, in terms of the lattice constant, a.  To put this in other words, if M is (0.5, 0) or (0, 0.5), and K is (1/3, 1/3), then what is the wavelength of the eigenstates in terms of the hexagonal lattice constant, a?  Or is there a quick way to use coordinates of a k-point to obtain the wavelength a 1x1 unit cell eigenstate would have at that point?

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General Questions and Answers / Re: How is a Mulliken population broken up into orbital contributions?

« on: June 28, 2010, 17:18 »

Nordland, thanks for the explanation.

Is it possible for a particular trace of the Mulliken population matrix with selections of specific angular momentum orbital sets to be a negative value?  If so, how is it achieved given that we have an electron density matrix and an orbital overlap matrix that we take the product of to compute the Mulliken population?  In some of the Mulliken population results I am studying, a few of the numbers are negative, while most of the other numbers are positive.  This is apparent in large basis sets, with lots of factors in their total sums.  The total populations, which are the sums of all the traces in different angular momentum sets, however, are positive.

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General Questions and Answers / How is a Mulliken population broken up into orbital contributions?

« on: June 27, 2010, 14:56 »

Friends,
     I have recently examined the effects of basis sets on Mulliken populations and other information in graphene and hexagonal BN layers.  When I keep the verbosity level at 10 in the NanoLanguage Scripter then watch the SCF convergence process, I notice that the Mulliken populations are distributed among orbitals of various angular momentum.  Parts of the population are allocated to s-orbitals, p-orbitals, and in the polarized basis sets, d-orbitals.  In the bilayers of graphene and h-BN, I observe that there are minor changes in some orbital contributions to the Mulliken population when the interface is formed, while other orbital shells do not change at all.

     How is this done, and is there any references related to decomposition of Mulliken populations into individual orbital factors?  This has never been extensively described in the literature, and I had considered looking at orbital contributions to try identify the major orbitals responsible for the interface that forms between the two different monolayers via pi-pi orbital overlap and hence the cause of energy perturbations that open the band gap in the graphene layer.

4

General Questions and Answers / Why use 'CalculateEffectivePotential'?

« on: June 21, 2010, 23:03 »

I recently computed contour plots of both graphene and hexagonal boron nitride monolayer in intralayer planes and planes perpendicular to the layers that bisect the bonds in these layers.  There are three of these plots for each plane type, and I am already able to understand the contour plots generated by 'CalculateElectrostaticDifference', which provides a plot similar to the molecular electrostatic potential (MEP) surface.  I also am able to interpret the plots produced by 'CalculateElectronDensity', which shows the electron density independent of orbitals in the substances.  However, I am not able to get adequate understanding of 'CalculateEffectivePotential', which gives the Kohn-Sham effective potential, V(eff), of the system.

What kind of utility is obtained by calculating contours of the Kohn-Sham effective potential?  I notice that the contour plots of 'effective potential' have values that never become positive, unlike the electron density plots (which have positive values).  How does one interpret the Kohn-Sham effective potential in terms of the atoms of a system?

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General Questions and Answers / Re: Agreement between Mulliken populations in theory and experiment?

« on: June 19, 2010, 04:19 »

Thanks, Dr. Blom.

I already have a similar figure in my dissertation that employs the electron density contour methods of VNL for the hexagonal BN bond, along with a C-C bond in the same unit cell for comparison.  We were able to infer how the density looks like, along with the electrostatic contours, in this structure.  Both helped us determine how the atoms were charged, as the electrostatic image gives us basins of attraction for a cationic probe.  The contours also gave us a way to begin our perturbation theory analysis of the C...X interactions, using the basins to define the energy shift directions for each interaction according to first-order and second-order perturbations.

Now I see about the arbitrary aspect...so it is based on particular definitions.  In that situation, we are seeing different definitions of the same quantity yielding different directions.  The positive thing about this is the magnitude...it is almost the same in both situations for us.  It shows that we got the ionicity of the B-N bond and potentially the atom sizes with a good estimate.

I'll take a look at the figures and see what I can do with the h-BN layer calculations.

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General Questions and Answers / Agreement between Mulliken populations in theory and experiment?

« on: June 18, 2010, 17:36 »

Fellows,
      I am here with a question about the method of Mulliken populations for atoms in a system.  I am asking it because calculations using LDA-PZ methods on hexagonal boron nitride leads to a Mulliken population of about 3.5 electrons for boron atoms and 4.5 electrons for nitrogen atoms.  This makes boron anionic and nitrogen cationic in these layers. However, in Journal of Physical Chemistry of Solids by Yamamura et al. (1997, volume 58, number 2, pages 177-183), the study of hexagonal BN with X-ray diffraction reveals a charge transfer FROM boron to nitrogen.  In the LDA-PZ results, the charge transfer is in the opposite direction, from nitrogen to boron.

How can we explain the discrepancy in the direction of charge transfer for hexagonal BN in these situations?

The LDA-PZ results correspond closely to what one would expect from a Lewis structure drawing of a h-BN layer, as double bond resonance would be needed to keep the layer planar and properly describe the bond length in this layer.  The bond length of B-N in the default model of h-BN is about 1.446 angstroms long.  Compare this with the typical B-N bond in borazine (1.436 angstroms), which is also described in the same way as h-BN layers.

Are there articles that describe agreement between Mulliken population calculations for materials and the experimental results obtained from them?

This is related to my current dissertation research, as the experimental results are not presenting the same picture of charge density in h-BN as our calculations.  I need to find out how to resolve this problem.

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General Questions and Answers / Re: Effects of spin polarization on unit cell with even number of electrons

« on: April 14, 2010, 17:11 »

Yes, I am aware of the polarization effects at defect sites. 

I have indeed found that the eigenstate occupancy in graphene does suggest spin pairing.  In the HOMO and HOMO-1 eigenstates, there are two electrons occupying them.  This is found in both spin-unpolarized and spin-polarized (with initial spin set to zero) calculations of graphene.  The electrons are remaining paired, despite the degeneracy of the initial states.

However, is this actually the experimentally observed microstate (two pairs of electrons in two stabilized orbitals and two empty orbitals, rather than four electrons residing in their own orbitals in an antiferromagnetic orientation with total spin equal to zero) for graphene?  Please find out if there is a reference that clearly states the microstate for graphene at K. 

It seems that the spin pairs would stabilize the orbitals they were located in, lowering those orbitals' energies.  In such a situation, the four-fold degeneracy would be lifted by the electron pairs, and we would not get the zero band gap in graphene at K.  How do we resolve this issue, given that graphene is observed to have a Dirac intersection point and degenerate eigenstates at this point?

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General Questions and Answers / Effects of spin polarization on unit cell with even number of electrons

« on: April 13, 2010, 17:13 »

Fellows,

I have been investigating graphene for a while, having finished a draft of my thesis paper on its band structure and how it can be modulated by atomic interactions.

When I read a recent Nature article, titled 'Quantum Spin Liquid emerging in two-dimensional correlated Dirac fermions' by Meng et al., I noted that their lattice was hexagonal, much like graphene.  This inspired me to try spin polarization on graphene, even though I had done calculations without using it.  As graphene has an even number of electrons in its orbitals, which are multiply-fold degenerate at K, we have configurations that can have fully paired up electrons in these states.  At K, the eigenstates are four-fold degenerate (two pairs of degenerate states, with different dispersion), meaning that we have four separate states to occupy with four of the eight available electrons for graphene.

This brings me to the question: What would be the effects of spin polarization on such a system?  How would the spin pairings be treated in it (unpolarized or polarized)?  I have observed slight differences at K when doing a spin-polarized calculation with initial spin set at zero.

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General Questions and Answers / Generalized gradient approximation

« on: March 11, 2010, 16:26 »

Hello, friends.

I have been continuing with my dissertation work-up and have come across some questions that I may wish to ask here.

When I was comparing local density approximation (LDA) with generalized gradient approximation (GGA), I noted that the difference between these Kohn-Sham methods is the way GGA takes in account the gradient of the density at a coordinate being computed.  LDA does not include this feature in its calculations of the same electron density, focusing only on the local electron density at the particular coordinate.  While I use both of these methods in my dissertation data, I want to know the benefits that inclusion of a gradient would have on the properties obtained from the electron density in a system (such as a unit cell and its contents).

Basically, how would gradient change the results of a calculation for the coordinate point?  How does it relate to the difference between LDA and GGA methods?

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General Questions and Answers / Re: Effect of supersize unit cell on band structure results

« on: January 14, 2010, 00:30 »

Thanks, Anders Blom.

I had discussed folding with my mentor when we were comparing the 1x1 unit cell with the 3x3 unit cell, which has the Dirac cone point at a different place.  I might bring in some pictures to show you the results for graphene done in both cells.  We had been using the 2x2 cells to examine the eigenstates found in the 1x1 cell models, so we decided to do a small set of models in this format.

In the pictures I provided, all graphene layers have been computed with Gamma-K-M-Gamma for the band structure pathway and good quality integration grids.  Each model has the original graphene unit cell repeated to create larger cells.  The band structures are for 1x1, 2x2, and 3x3.  You can clearly see where the Dirac cone is, by looking for the X-shape intersecting the Fermi energy level at zero energy.  Does zone folding explain why the Dirac cone shifts crystal momentum in the 3x3 cell?

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General Questions and Answers / Effect of supersize unit cell on band structure results

« on: January 13, 2010, 17:21 »

This is Quantamania with a new question.  I have been finishing up my dissertation work, and we are putting together results that were obtained using the graphene and h-BN layers in unit cells that are big enough to include 2x2 or more repetitions of atoms along the layers.  A 1x1 unit cell of these layers has 2 atoms and initial lattice dimensions, while a 2x2 unit cell has 8 atoms and double the lattice parameter length along the a-axis.

Here is my question: what are the effects of a 2x2 or larger unit cell on calculations of the resultant band structure originally computed for a 1x1 unit cell? 

So far, I have observed some significant changes compared to a 1x1 unit cell (the smallest cell needed for the models), even when using a 5x5x5 integration grid.  This will help me understand my new work and make connections between solid-state physics and quantum chemistry.

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General Questions and Answers / Re: Counting the atomic orbitals in a basis set

« on: December 10, 2009, 16:06 »

How do I find this information in the Virtual NanoLab or ATK file directory tree?  I currently have Virtual NanoLab 2008.10 on my computers.  Even though this does answer some of my questions, it does not fully help me in the case of carbon, boron and nitrogen.  These are the elements I only use in my dissertation studies, and will need the types of orbitals so I can describe them in my draft.

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General Questions and Answers / Counting the atomic orbitals in a basis set

« on: December 7, 2009, 17:00 »

Greetings.

I have a question for you:

I read one of the old threads about the basis set for Virtual NanoLab.  One of the responses described the atomic orbital components in hydrogen and gold for different basis sets.  I want to know how can we quickly determine the number of pseudopotential orbitals in a particular atom, based on its valence electron configuration.

In my instance, I have carbon, which has s and p orbitals available for describing the valence shell.  This is also true for boron and nitrogen.  How many UNIQUE atomic pseudopotential orbitals are in a carbon atom at SZ, SZP, DZ, DZP, and DZDP?  This will help me with my dissertation defense, particularly when discussing the number of primitives in the cells of my calculations.  It will also help me identify the polarization orbitals for these atoms.

Are there general formulas that work for counting basis functions in elements using valence shells only?

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General Questions and Answers / Re: Eigenstate occupancy arrays

« on: November 2, 2009, 17:16 »

Thanks for the reply.

In the example of graphene, I have an array of numbers at eigenstate #3 (its HOMO).  In this array, all of the numbers except for one, are 2.  The lone outlier is around 1.97 qe, and the band energy for this eigenstate is around 0.0001 eV below the Fermi energy.  Only at K (1/3, 1/3, 0) does this condition arise for graphene.  These results were obtained using an electron temperature of 300 K (for the Fermi-Dirac distribution).  With the other eigenstates that are occupied, I do not find an outlier k-point value anywhere.

What is the physical meaning of a single k-point showing significantly less eigenstate occupancy than other k-points adjacent to it?

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General Questions and Answers / Eigenstate occupancy arrays

« on: November 2, 2009, 15:35 »

How are the array data for eigenstate occupancy generated?  This choice only has one parameter and generates arrays for all of the Bloch eigenstates, including virtual ones.  In my results for graphene and mixed h-BN/graphene bilayers in supersized cells, I find that nearly all of the numbers in the occupied orbital arrays are 2, but one of the values differs significantly from all of the others (by as much as 0.025-0.03 qe).  This does not happen with monolayer h-BN, as it is an insulator.  Does this particular value have any meaning?