# Category Archives: Gaussian

## The ‘art’ of finding Transition States Part 1

Guillermo Caballero, a graduate student from this lab, has written this two-part post on the nuances to be considered when searching for transition states in the theoretical assessment of reaction mechanisms. He’s been quite successful in getting beautiful energy profiles for organic reaction mechanisms, some of which have even explained why some reactions do not occur! A paper in Tetrahedron has just been accepted but we’ll talk about it in another post. I wanted Guillermo to share his insight into this hard practice of computational chemistry so he wrote the following post. Enjoy!

Yes, finding a transition state (TS) can be one of the most challenging tasks in computational chemistry, it requires both a good choice of keywords in your route section and all of your chemical intuition as well. Herein I give you some good tricks when you have to find a transition state using Gaussian 09 Rev. D1

METHOD 1. The first option you should try is to use the opt=qst2 keyword. With this method you provide the structures of your reagents and your products, then the program uses the quadratic synchronous transit algorithm to find a possible transition state structure and then optimize it to a first order saddle point. Here is an example of the input file.

```link 0
--blank line--
#p b3lyp/6-31G(d,p) opt=qst2 geom=connectivity freq=noraman
--blank line--
Charge Multiplicity
Coordinates of reagents
--blank line--
Charge Multiplicity
Coordinates of products
--blank line---```

It is mandatory that the numbering must be the same in the reagents and the products otherwise the calculation will crash. To verify that the label for a given atom is the same in reagents and products you can go to Edit, then Connection. This opens a new window were you can manually modify the numbering scheme. I suggest you to work in a split window in gaussview so you can see at the same time your reagents and products.

The keyword freq=noraman is used to calculate the frequencies for your optimized structure, it is important because for a TS you must only observe one imaginary frequency, if not, then that is not a TS and you have to use another method. It also occurs that despite you find a first order saddle point, the imaginary frequency does not correspond to the bond forming or bond breaking in your TS, thus, you should use another method. I will give you advice later in the text for when this happens. When you use the noraman in this keyword you are not calculating the Raman frequencies, which for the purpose of a TS is unnecessary and saves computing time. Frequency analysis MUST be performed AT THE VERY SAME LEVEL OF THEORY at which the optimization is performed.

The main advantage for using the qst2 option is that if your calculation is going to crash, it generally crashes at the beginning, in the moment of guessing your transition state structure. Once the program have a guess, it starts the optimization. I suggest you to ask the algorithm to calculate the force constants once, this generally improves on the convergence, it will take slightly more time depending on the size of your structure but it pays off. The keyword in the route section is opt=(qst2,calcfc). Indeed, I hardly encourage you to use the calcfc keyword in any optimization you want to run.

METHOD 2. If method 1 does not work, my next advice is to use the opt=ts keyword. For this method, the coordinates in your input file are those for the TS structure. Here is an example of the input file.

```link 0
--blank line--
#p b3lyp/6-31G(d,p) opt=ts geom=connectivity freq=noraman
--blank line--
Charge Multiplicity
Coordinates of TS
--blank line--```

The question that arises here is how should I get the coordinates for my TS? Well, honestly this is not a trivial task, here is where you use all the chemistry you know. For example, you can start with the coordinates of your reagents and manually get them closer. If you are forming a bond whose length is to be 1.5Å, then I suggest you to have that length in 1.6Å in your TS. Sometimes this becomes trial and error but the most accurate your TS structure is, based on your chemical knowledge, the easiest to find your TS will be. As another example, if you want to find a TS for a [1,5]-sigmatropic reaction a good TS structure will be putting the hydrogen atom that migrates in the middle point through the way. I have to insist, this method hardly depends on your imagination to elucidate a TS and on your chemistry background.

Most of the time when you use the opt=ts keyword the calculations crashes because of an error in the number of eigenvalues, you can avoid it adding noeigen to the route section; here is an example of the input file, I encourage you to use this method.

```link 0
--blank line--
#p b3lyp/6-31G(d,p) opt=(ts,noeigen,calcfc) geom=connectivity freq=noraman
--blank line--
Charge Multiplicity
Coordinates of TS
--blank line--```

If you have problems in the optimization steps I suggest you to ask the algorithm to calculate the force constants in every step of the optimization opt=(ts,noeigen,calcall) this is quite a harsh method because will consume long computing time but works well for small molecules and for complicated TSs to find.

Another ‘tricky’ way to get your coordinates for your TS is to run the qst2 calculation, then if it fails, take the second- or the third-step coordinates and used them as a ‘pre-optimized’ set of coordinates for this method.

By the way, here is another useful trick. If you are evaluating a group of TSs, let’s say, if you are varying a functional group among the group, focus on finding the TS for the simplest case, then use this optimized TS as a template where you add the moieties and use this this method. This works pretty well.

For this post we’ll leave it up to here and post the rest of Guillermo’s tricks and advice on finding TS structures next week when we’ll also discuss the use of IRC calculations and some considerations on energy corrections when plotting the full energy profile. In the mean time please take the time to rate, like and share this and other posts.

## Partial Optimizations with Gaussian09

Sometimes you just need to optimize some fragment or moiety of your molecule for a number of reasons -whether because of its size, your current interest, or to skew the progress of a previous optimization- or maybe you want just some kind of atoms to have their positions optimized. I usually optimize hydrogen atoms when working with crystallographic files but that for some reason I want to preserve the rest of the molecule as refined, in order to keep it under a crystalline field of sorts.
Asking Gaussian to optimize some of the atoms in your molecule requires you to make a list albeit the logic behind it is not quite straightforward to me. This list is invoked by the ReadOptimize keyword in the route section and it includes all atoms by default, you can then further tell G09 which atoms are to be included or excluded from the optimization.
So, for example you want to optimize all atoms EXCEPT hydrogens, then your input should bear the ReadOptimize keyword in the route section and then, at the end of the molecule specification, the following line:

`atoms notatoms=H`

If you wish to selectively add some atoms to the list while excluding others, here’s an example:

`atoms=C H S notatoms=5-8`

This list adds, and therefore optimizes, all carbon, hydrogen and sulfur atoms, except atoms 5, 6, 7 and 8, should they be any of the previous elements in the C H S list.
The way I selectively optimize hydrogen atoms is by erasing all atoms from the list -using the noatoms instruction- and then selecting which are to be included in the list -with atoms=H-, but I haven’t tried it with only selecting hydrogen atoms from the start, as in atoms=H

`noatoms atoms=H`

I probably get very confused because I learned to do this with the now obsolete ReadFreeze keyword; now it sometimes may seem to me like I’m using double negatives or something – please do not optimize all atoms except if they are hydrogen atoms. You can include numbers, ranks or symbols in this list as a final line of your input file.

Common errors (by common I mean I’ve got them):

Lets look at the end of an input I just was working with:

>  AtmSel:  Line=”P  0″
>  Maximum list size exceeded in AddBin.
>  Error termination via Lnk1e in…

AtmSel is the routine which reads the atoms list and I was using a pseudopotential on phosphorous atoms, I placed the atoms list at the end of the file but it should be placed right after the coordinates and the connectivity matrix, should there be one, and thus before any external basis set or pseudopotential or any other specification to be read by Gaussian.

As a sort of test you can use the instruction:

```%kjob l103
%chk=myfile.chk
...```

at the Link0 section (where your checkpoint is defined). This will kill the job after the link 103 is finished, thus you will only get a list of what parameters were frozen and which were active. Then, if things look ok, you can run the job without the %kjob l103 instruction and get it done.

As usual I hope this helps. Thanks for reading except to those who didn’t read it except for the parts they did read.

## Efficient use of the clipboard on GaussView

Editing large molecules on a seemingly simple visualizer as GaussView can be a bit daunting. I’m working on a follow up of that project we recently published in JACS but now we require to attach two macrocycles to the organometallic moiety; the only caveat is that this time we don’t have any crystallographic data with which to start. Generating a 3D model of this structure is already hard enough and even when you managed to do it there are many degrees of freedom that in some cases can lead to unrealistic geometries after optimization.

I recently came across a simple way to edit a large complicated molecule by optimizing the fragments separately and then joining them in a new molecule by using the clipboard. This rather simple method, that I for one had never exploited has just saved me a few good hours.

Copy a molecule (CTRL+C) and it will go to the clipboard as a molecular fragment for which you can define a new hot atom and thus bind it to the other fragment as you would with the regular builder. I strongly suggest to use a “New Molecule Group” instead of editing over an existing molecule. Also, if you are using the “paste” button, observe that it has three different options; you may want to use the last one “append to existing molecule” or you will have your fragments in different windows.

And remember, dihedral angles are your best friends.

## Atoms in Molecules (QTAIM) – Flash lesson

As far as population analysis methods goes, the Quantum Theory of Atoms in Molecules (QTAIM) a.k.a Atoms in Molecules (AIM) has become a popular option for defining atomic properties in molecular systems, however, its calculation is a bit tricky and maybe not as straightforward as Mulliken’s or NBO.

Personally I find AIM a philosophical question since, after the introduction of the molecule concept by Stanislao Cannizzaro in 1860 (although previously developed by Amadeo Avogadro who was dead at the time of the Karlsruhe congress), the questions of whether or not an atom retains its identity when bound to others? where does an atom end and the next begins? What are the connections between atoms in a molecule? are truly interesting and far deeper than we usually consider because it takes a big mental leap to think about how matter is organized to give rise to substances. Particularly I’m very interested with the concept of a Molecular Graph which in turn is concerned with the way we “draw lines” to form conceptual molecules. Perhaps in a different post we can go into the detail of the method, which is based in the Laplacian operator of the electron density, but today, I just want to collect the basic steps in getting the most basic AIM answers for any given molecule. Recently, my good friend Pezhman Zarabadi-Poor and I have used rather extensively the following procedure. We hope to have a couple of manuscripts published later on. Therefore, I’ve asked Pezhman to write a sort of guest post on how to run AIMALL, which is our selected program for the integration algorithm.

The first thing we need is a WFN or WFX file, which contains the wavefunction in a Fortran unformatted file on which the Laplacian integration is to be performed. This is achieved in Gaussian09 by incluiding the keyword output=wfn or output=wfx in the route section and adding a name for this file at the bottom line of the input file, e.g.

`filename.wfn`

(NOTE: WFX is an eXtended version of  WFN; particularly necessary when using pseudopotentials or ECP’s)

Analyzing this file requires the use of a third party software such as AIMALL suite of programs, of which the standard version is free of charge upon registration to their website.

OpenAIMStudio (the accompanying graphical interface) and select the AIMQB program from the run menu as shown in figure 1.

Figure 1

Select your WFN/WFX file on which the calculation is to be run. (Figure 2)

Figure 2

You can control several options for the integration of the Laplacian of the electron density as well as other features. If your molecules are simple enough, you may go through with a successful and meaningful calculation using the default settings. After the calculation is finished, several result files are obtained. We’ll work in this tutorial only with *.mpgviz (which contains information about the molecular graph, MG) and *.sum (which contains all of  needed numerical data).

Visualization of the MG yields different kinds of critical points, such as: 1) Nuclear Attractor Critical Points (NACP); 2) Bond Critical Points (BCP); 3) Ring CP’s (RCP); and 4) Cage CP’s (CCP).

Of the above, BCP are the ones that indicate the presence of a chemical bond between two atoms, although this conclusion is not without controversy as pointed out by Foroutan-Njead in his paper: C. Foroutan-Nejad, S. Shahbazian and R. Marek, Chemistry – A European Journal, 2014, 20, 10140-10152. However, at a first approximation, BCP’s can help us to explore chemical interactions.

Now, let’s go back to visualizing those MGs (in our examples we’ve used methane and ethylene and acetylene). We open the corresponding *.mpgviz file in AIMStudio and export the image from the file menu and using the save as picture option (figure 3).

Figure 3

The labeled atoms are NACP’s while the green dots correspond to BCP’s. Multiplicity of a bond cannot be discerned within the MG; in order to find out whether a bond is a single, double or triple bond we have to look into the *.sum file, in which we’ll take a look at the bond orders between pairs of atoms in the section labeled “Diatomic Electron Pair Contributions and Delocalization Data” (Figure 4).

Figure 4

Delocalization indexes, DI’s, show the approximate number of electrons shared between two atoms. From the above examples we get the following DI(C,C) values: 1.93 for C2H4 and 2.87 for C2H2; on the other hand, DI(C,H) values are  0.98 for CH4, 0.97 in C2H4 and 0.96 in C2H2. These are our usual bond orders.

This is the first part of a crash tutorial on AIM, in my opinion this is the very basics anyone needs to get started with this interesting and widespread method. Thanks to all who asked about QTAIM, now you have your long answer.

Thanks a lot to my good friend Dr Pezhman Zarabadi-Poor for providing this contribution to the blog, we hope you all find it helpful. Please share and comment.

## Some .fchk files wont open in GaussView5.0 (Update)

A couple of weeks ago I posted a solution for a common error regarding .fchk files that will display the error below when opened with GaussView5.0. As I expected, this error has to do with the use of diffuse functions in the basis set and is related to a change of format between Gaussian versions.

```CConnectionGFCHK::Parse_GFCHK()
Missing or bad data: Alpha Orbital Energies
Line Number 1234```

Although the method described in the previous post works just fine, the following update is a better approach. Due to a change of spelling between G03 and G09 (which has been corrected for G09 but not available for GV versions prior to 5.0.9) one must change “independent” for “independant

To make the change directly from the terminal the following command is needed:

`sed -i 's/independent/independant/g' file.fchk`

Alternatively you can redirect the output to a new file

`sed -e 's/independent/independant/g' file.fchk > newfile.fchk`

if you want to keep the old version and work with a new one.

Of course this edition can be performed manually with any text editor available (for example if you work in Windows) but solutions from the terminal always seem easier and a lot more fun to me.

Thanks to Dr. Fernando Cortés for sharing his insight into this issue.

## If a .fchk file wont open in GaussView5.0

I’ve found the following error regarding the opening of .fchk files in GaussView5.0.

```CConnectionGFCHK::Parse_GFCHK()
Missing or bad data: Alpha Orbital Energies
Line Number 1234```

The error is prevented to a first approximation (i.e. it at least will allow GV to open and visualize the file but other issues may arise) by opening the file and modifying the number of basis functions to equal the number of independent functions (which is lower)

```FILE HEADER
FOpt RM062X 6-311++G(d,p)
Number of atoms I 75
Info1-9 I N= 9
163 163 0 0 0 110
2 18 -502
Charge I 0
Multiplicity I 1
Number of electrons I 314
Number of alpha electrons I 157
Number of beta electrons I 157
Number of basis functions I 1199
Number of independent functions I 1199
Number of point charges in /Mol/ I 0
Number of translation vectors I 0
Atomic numbers I N= 75
... ...
... ...```

Once both numbers match you can open the file normally and work with it. My guess is this will continue to happen with highly polarized basis sets but I need to run some tests.

## Transition State Search (QST2 & QST3) and IRC with Gaussian09

Theoretical evaluation of a reaction mechanism is all about finding the right transition states (TS) but there are no guarantees within the available methods to actually find the one we need. Chemical intuition in the proposal of a mechanism is paramount. Let’s remember that a TS is a critical point on a Potential Energy Surface (PES) that is a minimum in every dimension but one. For a PES with more than two degrees of freedom, a hyper-surface, envisioning the location of a TS is a bit tricky, in the case of a three dimensional PES (two degrees of freedom) the saddle point constitutes the location of the TS as depicted in figure 1 by a section of a revolution hyperboloid.

Fig1. Saddle point on a surface (min in one direction; max in the other)

Fig 1a Pringles chips -Yuck-. They exhibit a maximum on the direction parallel to the screen and a minimum on the direction perpendicular to the screen at the same point.

The following procedure considers gas phase calculations. Nevertheless, the use of the SCRF keyword activates the implicit solvent calculation of choice in order to evaluate to some degree the solvent influence on the reaction energetics at different temperatures with the use of the temperature keyword.

The first step consists of a high level optimization of all minimums involved, such as reagents, products and intermediates, with a subsequent frequency analysis that includes no imaginary eigenvalues.

In order to find the structures of the transition states we use in Gaussian the Synchronous Transit-guided Quasi-Newton method [1] through the keywords QST2 or QST3. In the former case, coordinates for the reagents and products are needed as input; for the latter keyword, coordinates for the TS structure guess is needed also.

QST2)

%chk=file.chk
%nprocshared=n
%mem=nGB

#p opt=(qst2,redundant) m062x/6-31++G(d,p) freq=noraman Temperature=373.15 SCRF=(Solvent=Water)

Title card for reagents

Q M
Cartesian Coordinates for reagents
blank line
Title card for products

Q M
Cartesian Coordinates for products
blank line

QST3)

%chk=file.chk
%nprocshared=n
%mem=nGB

#p opt=(qst3,redundant) m062x/6-31++G(d,p) freq=noraman Temperature=373.15 SCRF=(Solvent=Water)

Title Card for reagents

Q M
Cartesian Coordinates for reagents
blank line
Title card for products

Q M
Cartesian Coordinates for products
blank line–
Title card for TS
Q M
Cartesian Coordinates for TS
blank line

NOTE: It is fundamental that the numbering order is kept constant throughout the molecular specifications of all two, or three, input structures. Hence, I recommend to build a set of molecules, save their structure, and then modified the coordinates on the same file to produce the following structure, that way the number for every atom will remain the same for every step.

As I wrote above, there are no guarantees of finding the right TS so many attempts are probably needed. Once we have the optimized structures for all the species involved in our mechanistic proposal we can plot their energies very simply with MS Excel the way we’ve previously described in this blog (reblogged from eutactic.wordpress.com)

Once we’ve succeeded in finding the structure of our TS we may run an Internal Reaction Coordinate (IRC) calculation. This calculation will connect the TS structure to those of the products and the reagents. Initial constant forces are required and these are commonly retrieved from the TS calculation checkpoint file through the RCFC keyword.

%chk=QST3_2p.chk
%nprocshared=8

#p m062x/6-31++G(d,p) IRC=(Maxpoints=50,RCFC,phase=(2,1))Temperature=373.15 SCRF=(Solvent=Water) geom=allcheck

Title Card

Q M
blank line

Finally, the IRC path can be visualized with GaussView from the Results menu. A successful IRC will link both structures along a single reaction coordinate proving that both reagents and products are linked by the obtained TS.

Hat tip to Howard Diaz who has become quite skillful in calculating these mechanisms as proven by his recent poster at the XII RMFQT a couple of weeks back. And as usual thanks to everyone who reads, comments, likes, recommends, rates and shares my silly little posts.

## XIIth Mexican Reunion on Theoretical Physical Chemistry

As every year this month we had the yearly Mexican Reunion on Theoretical Physical Chemistry organized by prominent researchers in the field, such as Dr. Emilio Orgaz (UNAM), Dr. Alberto Vela (CINVESTAV) and many other. Over 150 different works were presented during this edition which took place in Juriquilla, Querétaro at one of the many campuses of the National Autonomous University of Mexico scattered all around the country. Below you can see some pictures from the talks and the first poster session.

This time we contributed with a small poster on a mechanism proposed by Howard Diaz (an undergrad student from UAEM) on the equilibrium transformation of dihydrocinolines into 1-amino-indoles by an intramolecular rearrangement. May this post also serve as the starting point of a -mini-tutorial on how to evaluate a mechanism theoretically using QST3 and IRC in implicitly solvated environments (PCM)

Howard Diaz posing next to his poster

The equilibrium under study and the proposed mechanism  by which it occurs, originally proposed by Frontana-Uribe et al. looks a bit like this:

Dihydrocinolines in equilibrium with 1-aminoindole

Mechanistic proposal by Frontana-Uribe et al.

The energy profile, in which all transition states were calculated with the QST3 method, is presented below, calculated at various levels of theory. Also, the Internal Reaction Coordinate (IRC) connecting both states was calculated and is shown further below in the full poster.

Energy Profile

From this results we believe that a new mechanistic proposal is needed since the energy barrier for the first step is quite high (~60 kcal/mol) and hence a bit unlikely to occur through that transition state. Nevertheless this is a first approach to elucidating a mechanism and the more knowledge about it the higher the control will be on this chemical transformation.

A full version of the poster is shown below for your convenience (Spanish). See you all at the next RMFQT in Morelia 2014!

Full Poster

## Delta G of solvation in Gaussian09

How to calculate the Delta G of solvation? This is a question that I get a lot in this blog, so it is about time I wrote a (mini)post on it, and at the same time put an end to this posting drought which has lasted for quite a few months due to a lot of pending work with which I’ve had to catch up. Therefore, this is another post in the series of SCRF calculations that are so popular in this blog. For the other posts on this subjects remember to click here and here.

SMD

SMD is the keyword you want to use when performing a Self Consistent Reaction Field (SCRF) calculation with G09. This keyword was only made available in this last version of the program and it corresponds to Truhlar’s and coworkers solvation model which is recommended by Gaussian itself as the preferred model to calculate Delta G of solvation. The syntax used is the standard way used in any other Gaussian input files as follows:

`# 'route section keywords' SCRF=SMD`

Separately, we must either perform a gas phase calculation or use the DoVacuum keyword within the same SCRF input, and then take the energy difference between gas phase and solvated models.

`# 'route section keywords' SCRF=(SMD,DoVacuum)`

No solvation or cavity model should be defined since, by definition, SMD will use the IEFPCM model which is a synonym for PCM.

As opposed to the previous versions of Gaussian, the output energy already contains all corrections, this is why we must take the difference between both values (remember to calculate them both at the same level of theory if calculated separately!). Nevertheless, when using the SMD keyword we get a separate line, just below the energy, stating the SMD-CDS non electrostatic value in kCal/mol.

The radii were also defined in the original paper by Truhlar; I’m not sure if using the keyword RADII with any of its options yields a different result or if it even ends in an error. Its worth the try!

Some calculation variations are not available when using SMD, such as Dis (calculation of the solute-solvent dispersion interaction energy), Rep (solute-solvent repulsion interaction energy) and Cav (inclusion of the solute cavitation energy in the total energy). I guess the reason for this might be that the SMD model is highly parametrized.

Have you found any issue with any item listed above? Pleases share your thoughts in the comments section below. As usual I hope this post was useful and that you all rate it, like it and comment.

References

A. V. Marenich, C. J. Cramer, and D. G. Truhlar, “Universal solvation model based on solute electron density and a continuum model of the solvent defined by the bulk dielectric constant and atomic surface tensions,” J. Phys. Chem. B, 113 (2009) 6378-96.

## Polarizability and Hyperpolarizability in Gaussian

Calculating both Polarizability and the Hyperpolarizability in Gaussian is actually very easy and straightforward. However, interpreting the results requires a deeper understanding of the underlying physics of such phenomena. Herein I will try to describe the most common procedures for calculating both quantities in Gaussian09 and the way to interpret the results; if possible I will also try to address some of the most usual problems associated with their calculation.

Introduction

The dipole moment of a molecule changes when is placed under a static electric field, and this change can be calculated as

pe = pe,0 + α:E + (1/2) β:EE + … (1)

where pe,0 is the dipole moment in the absence of an electric field; α is a second rank tensor called the polarizability tensor and β is the first in an infinite series of dipole hiperpolarizabilities. The molecular potential energy changes as well with the influence of an external field in the following way

U = U0 – pe.E – (1/2) α:EE – (1/6) β:EEE – … (2)

.

Route Section Keyword: Polar

This keyword requests calculation of the polarizability and, if available, hyperpolarizability for the molecule under study. This keyword is both available for DFT and HF methods. Hyperpolarizabilities are NOT available for methods that lack analytic derivatives, for example CCSD(T), QCISD, MP4 and other post Hartree-Fock methods.

Frequency dependent polarizabilities may be calculated by including CPHF=RdFreq in the route section and then specifying the frequency (expressed in Hartrees!!!) to which the calculation should be performed, after the molecule specification preceded by a blank line. Example:

```#HF/6-31G(d) Polar CPHF=RdFreq

Title Section

Charge Multiplicity
Molecular coordinates
==blank line==
0.15```

In this example 0.15 is the frequency in Hartrees to which the calculation is to be performed. By default the output file will also include the static calculation, that is, ω = 0.0. Below you can find an example of the output when the CPHF=RdFreq is employed (taken from Gaussian’s website) Notice that the second section is performed at ω = 0.1 Ha

``` SCF Polarizability for W=    0.000000:
1             2             3
1  0.482729D+01
2  0.000000D+00  0.112001D+02
3  0.000000D+00  0.000000D+00  0.165696D+02
Isotropic polarizability for W=    0.000000       10.87 Bohr**3.
SCF Polarizability for W=    0.100000:
1             2             3
1  0.491893D+01
2  0.000000D+00  0.115663D+02
3  0.000000D+00  0.000000D+00  0.171826D+02

Isotropic polarizability for W=    0.100000       11.22 Bohr**3.```

You may have noticed now that the polarizabilities are expressed in volume units (Bohr^3) and the reason is the following:

Consider the simplest case of an atom with nuclear charge Q, radius r, and subjected to an electric field, E, which creates a force QE, and displaces the nucleus by a distance d. According to Gauss’ law this latter force is given by:

(dQ^2)/(4πεr^3) = QE      (Hey! WordPress! I could really use an equation editor in here!)

if the polarizability is defined by Qd/E then we can rearrange the previous equation and yield

α = 4πεr^3 which in atomic units yields volume units, r^3, since 4πε = 1. This is why polarizabilities are usually referred to as ‘polarizability volumes’.

****THIS POST IS STILL IN PROGRESS. WILL COMPLETE IT IN SHORT. SORRY FOR ANY INCONVENIENCE****

The use of double zeta quality basis sets is paramount but it also makes these calculations more time consuming. Polarization functions on the basis set functions are a requirement for good results.

As usual, please rate/comment/share this post if you found it useful or if you think someone else might find it useful. Thanks for reading!