Anyways, so I had a great time at GaTech meeting Dr. Sherrill! However, in the end I felt that UGA at the Center for Computational Quantum Chemistry is the best place for me to grow as a scientist, which is in the end what I'm going to grad school for right. Also, I'm still not sure whether I'm ready to live in the big city and I prefer quieter environments with lots of Southern cooking available(grits, meatloaf, fried okra, fried everything, ....). Finally, the pay at UGA doesn't hurt and cost of living.P.S.: If you have no idea what I've been talking about for most of this note, then the short of it is that I'm going to UGA to work at the CCQC starting June 2. Here are some links to those interested:

http://www.psicode.org

http://www.q-chem.com

http://www.ccc.uga.edu

1) To write a Hartree-Fock program(calculate the energy of a Lithium hydride) in FORTRAN, then port this to a GPGPU. 2) Calculate the energy of C6 structures and determine their quadropole moments. 3) Make a model of an atom-like system with just a dipole, and a system with just a quadrupole. Then, adjust the charge(q) and the distance seperating the charges(a) to determine how large a quadrupole needs to be to bind an electron, and use the dipole model to confirm that the model is sane.

A brief overview:

Computer simulations and visualizations are performing the thought experiments of the 21st century and pushing the limits of human vision and imagination.

Possibly related:

• New Architectures for a New Biology (by David E. Shaw)

• The Inner Life of the Cell

• Simulating Physics with Computers

What is the interesting point? As a simulator program, this software can calculate the set of reaction rates in very quick and acurate. The input file will be two types: first, use the Arrhenius model which your input data contains A (constanta of collosion), E (activation energy) and n (the order of reaction to the temperature) or the second just use the k (constant of reaction rate). If the reaction is in the balance condition (means it has forward and backward rate) you can choose the mode of balance and put the value of forward as well as the backward reaction. The input data processing is done one by one for each reaction, unfortunatelly, I don;t know whether there is a program to put the input with many reactions simultaneously. Anyway,to input the data is quite easy because it is already in graphical user interface window.

 Based on my experience, the calculation is quite very fast. The result can be exported in .cvs format so you can get the numerical results. For me, this program is very useful to calculate my process, e.i. plasma reaction, which can not be done by temperature- based kinetic program, such as Kintecus or Chemkin. In plasma process, especially non-thermal plasma, the tempearature is very low which not sufficient to produce numerous ions or radicals follows the Boltzmann distribution. In reality, the abundant reactive species were exsisted in the plasma which means there is a parameter which become a driving force insted of high temperature. Using this program, I can 'manipulate' the existance of electron and its rate. The elecron can be a major factor during the reactions of plasma.

Anyway, this program is useful. As it is also free, there is not a mistake to try and learn how to use even totally I don;t have any idea about the computational program or algorithm to solve the mathematical parts of solving the differential equation.

Compare to the previous version of Kintecus, the current version is having more advantages. One of them is sensitivity analysis which is useful to filter the only important reactions occured in the global process. The command is also quite easy for using this software, for example, just put keyword: -SENSIT:1 and the program will give you a set of sensitivity calculation in each time spant. Other advantages? ofcourse as kinetic program, this software will give you the trend of species. Before doing simulation, you have to put your calculation models in one sheet named model. In this sheet, you put all of your proposed reactions or if you don't have any idea about the reaction, just put as many as you get from other sources, for example chemkin software or GRI-Mech. The important is all reaction is order one to each reactant or product. All value of the reaction rate follows modified Arhennius equation. One good thing from this software is it considers the thermodynamic term. All properties of molecules which is has similar format with chemkin database .dat file. Even, if you are missing a molecule property, you can download or get this data from chemkin or GRI-Mech database. Just do googling to find it, it believe you can find it easily as my experience.

As the software is shareware for 10 days and you can obtain easy registration, I believe you can use it as your powerful tool to calculate your chemical reactions. The matter is just, please, spend around 20-30 minutes to understand it by read the manual. Once you know it, you will easily operate this software. The official kintecus website address is http://www.kintecus.com.

First, at the risk of sounding needlessly didactic, the frequencies are not "negative," they are imaginary. It is the computed force constant that is negative. In the quantum mechanical harmonic oscillator (QMHO) approximation, the frequency is related to the square root of the force constant, and hence it is imaginary. It is a historical curiosity, probably relating to the difference between FORTRAN floating point and character variables, that most codes print the specific frequencies as negative numbers rather than appending Euler's "i" after the magnitude.

On a more substantial front, several posters have offered suggestions for removing the "minor" imaginary frequency by perturbing geometries along the predicted normal mode, reoptimizing, generally jiggling structures, etc. Such procedures can indeed be effective, but only if the imaginary frequency is really there... Left unaddressed (at least in the most recent iteration of this thread -- I have vague memories that others may have posted to CCL on this point before) is the possibility that the unwanted imaginary frequency is an artifact of the quadrature grid used in a DFT calculation (recent posters have not actually specified their level of electronic structure theory, but given the prevalence of DFT in modern calculations, one suspects this was indeed their choice).

Most (if not all) modern DFT functionals do NOT permit an analytic evaluation of the necessary volume integrals of the exchange-correlation potential. Instead, the integrals are solved via a quadrature procedure over a 3-dimensional grid. The so-called analytic derivatives and second derivatives are thus NOT analytic derivatives of the correct integrals, they are analytic derivatives of the quadrature schemes for these integrals. The accuracy of any quadrature approach depends on the density of the grid points, and, of course, the cost of the integral evaluation also goes up with the number of grid points. So, codes like Gaussian, Jaguar, ADF, NWChem, ORCA, etc. (sorry if I left out your favorite) have come up with default integration grids that represent good compromises for speed and accuracy. However, the demands on grid size increase as one becomes interested in not just the value of an integral, but also in the value of its first and second derivatives with respect to atomic positions. Thus, it is not at all uncommon with default grids to find a geometry that seems by all accounts to be, say, a local minimum, but gives a small imaginary frequency even though it has no symmetry and all attempts to rotate methyl groups (for instance) fail to eliminate the problem. TS structures can certainly suffer from exactly the same phenomenon. The problem is that the force constant is not being computed accurately enough by the quadrature scheme, NOT that there really is a negative curvature on the potential energy surface.

So, what can one do? Some codes permit one to choose a finer quadrature grid, and this often does solve the problem. Of course, one can then worry about whether one should go back and recompute one's full set of stationary points with this finer grid (it's only computer time...) but at least one knows that the issue is not a chemical one. Another option is to compute the frequencies by finite difference of the first derivatives (probably even MORE expensive and not really an ideal option). Lastly, one can boldly rely on one's chemical intuition to know when one is being plagued by this problem and attempt to sleep well while blithely ignoring the issue (after all, the mode in question will have a "true" frequency that will contribute negligibly to zero-point vibrational energy, and negligibly to enthalpy, and will be so small that the correct value should not be used on the QMHO approximation for entropy in any case).

For those wishing for a more complete discussion of these issues, I believe that Fritz Schaefer and co-workers recently published a paper or two comparing the convergence of default quadrature grids for various properties in various codes, but I'm too lazy to do the literature search for these inquiring individuals. Happy hunting.

It is not easy to find the TS molecule. Very common, you will get a list of frequencies with two negative values, means double saddle point. This is meaningless as it has any chemical meaning. Or, one of them is very small value but existed. Also, sometime we confused whether some reactions have TS or not for example the homolitic breaking bond of CH4 into CH3 and H. The best way to do it is by scan the energy as a function of H distance to the C atom. If there is a curve profile, you can use the lowest value as the guess for TS search. Sometimes, it does not show the thing that we imagine. For example, I tried to add O2 to CH3 to see the probablitiy of having H3COO. As it is existed, I hope to get the TS. However, the lowest value of scan giving me the TS of C-O rotation, not the TS addition itself.

The basic meaning of ONIOM is computing the molecule with different level of theory simulataneously. The reason could be very simple, when you have a large molecule, it will spend a lot of time to calculate all parts with very precise method although you will get the 'almost' correct value. On the other hand, using low basis sets, you will produce untrustable result.  So, in the middle, you can manage the important part of molecule, doing 'more' in term of calculation, and use the low level theory for other parts which don't need to be very correct. As nowadays there are many basis sets for specific cases, the choice of that is the key of the correctness of the calculation. It requires many literature reviews before doing this case.

In ONION, you can split your molecule into three different levels of QC/MM or QC/QC. QC stands for quantum chemical while MM stands for molecular mechanical.  You can find this definition in this blog or by googling I hope you will know it. The definition of these three levels of calculation will be mentioned after you defind the geometry of single atom in the input file of Gaussian. Most methods works with Oniom, e.g. optimisations and frequency calculations. However, SCRF does not work (it is not available for semiempirical or molecular mechanical methods). SCRF or self-consistent reaction field is usually used for the calcilation with solvation (solvent existence) or non gaseous-phase.

The example of ONIOM input in the Gaussian is:

#P Oniom(B3LYP/6-31G*:PM3:Amber) Opt Freq

A three-layer oniom calculation of CH3-CH2-CHO

0 1
 h   1.303312051   0.000000000   1.949234062 High
 c   0.375760127   0.000000000   1.351034669 High
 o  -0.718742386   0.000000000   1.904192435 High
 c   0.583631613   0.000000000  -0.143478503 Medium H
 h   1.171836119  -0.887154331  -0.398970957 Medium
 h   1.171836119   0.887154331  -0.398970957 Medium
 c  -0.720914899   0.000000000  -0.916455389 Low H
 h  -1.320117994   0.884637788  -0.676628009 Low
 h  -1.320117994  -0.884637788  -0.676628009 Low
 h  -0.526482757   0.000000000  -1.993329341 Low

Multi-Coordinate Driven Reaction Path Search

MCD 1.2 by Imre Berente

I. Introduction

This program is to calculate approximate reaction path and transition state candidates for chemical reactions. For scientific details see article J. Phys. Chem. A 110 (2006) 772. If you use this program in your scientific work creating a publication, please refer to this article.

The program is a driver for Gaussian 03, which makes the actual calculations. The Gaussian executable g03.exe is needed in the path.

The program is available as WIN32 and Linux executable and also as source code. You are free to examine, modify or take subroutines from it as long as you refer to this site in your readme. The compilations were made by Freepascal compiler.

The test calculations of the article are also available. You can examine them as input examples.

The results of this program are NEVER accurate intermediates/transition states, their structures and energies should not be used for any other purpose than getting a qualitative picture and as initial guess geometries for optimizations of the intermediates and saddle point searches for the TSs.

Than why use this program?
Because it is robust, deals with any reaction and you can test even the fanciest reaction mechanism theory. Compaired with the other robust methods it is pretty fast. It provides you good initial guess geometries for the geometry optimizations and saddle-point searches which are hard to create manually.

Requirements

The program needs the optimized geometry of the reactants and also the products. The user also must have an idea about the mechanism he wants to calculate with this program. Based on this mechanism internal coordinates needed to be selected which change monotonously with the reaction coordinate (active coordinates).

Active coordinate selection

You shall select all "important" coordinates of the reaction. The bond lengths of all forming and broken bond. The torsion angle of a rotation which inherently belongs to the reaction. Selecting too few coordinates can make the program fail.

Faliures because of lack of coordinates can be spotted by examining the Energy vs reaction coordinate plot created from the mcd.tbl file. You should see gauss-like peaks. If you see a "left sided peak" like on the image here, you can assume that an important coordinate is missing. The program lists the nonactive coordinate changes in the output, if it is much bigger than the stepsize, you can expect such error. The missing active coordinate is often between the jumping ones. In rare cases it is possible that the peak cannot be removed by including new coordinates to the active subspace. In such case the method is not able to solve the problem.

Defining a coordinate active which does not changes monotonously with the reaction coordinate can make artificial results and must be avoided.

Defining and "uninportant" coordinate which changes monotonously with the reaction coordinate increase computational cost but should not cause any faliures.

The program in brief

During the calculation, the program will do the following steps:

1. Examine the forces and the Hessian on the active coordinates

2. The program change these coordinates. The sum of the changes is the given stepsize and the changes are distributed among the active coordinates in order to archive minimal energy increase.

3. The program runs the Gaussian to optimize all nonactive coordinates.

4. Unless all coordinates reached their ending value, GOTO 1

Input files

The program does not use parameters, all input are in the input files.

• mcd.par: the parameter file for the MCD program.

• filename.chk: the checkpoint file of the optimized input structure. MCD will take the input geometry and the initial Hessian from this file. If you have an optimized reactant geometry with no checkpoint, you shall run an optimization which will converge in a few steps.

Output files

• stepXXXX.gjf: input file for Gaussian to order the geometry optimization in the XXX-th step

• stepXXXX.out/log: output file from the Gaussian containing the energy and the geometry in the XXX-th step

• chksavXXX.chk: Gaussian checkpoint file containing lot of data from the XXX-th step. It is needed by the program for restarting. When finished, these huge files can be deleted.

• mcd.out: the same file which is displayed on the screen during run. Usually unimportant, maybe some warnings should be checked.

• mcd.tbl: summary of the calculation, contains the relative energies of the steps and also the values of the active coordinates.

How to check the result?

While the program runs it create steps. These steps are the milestones of the reaction path. You can load the step output files into GaussView and see if they are the ones you expected. When the run is complete, the last out file can be converged into the product geometry. If it is not and the difference is significiant than something is odd. Maybe the input was mistyped, the stepsize was too high, an important coordinate was not set active, maybe the reaction mechanism theory was bad. Using smaller stepsize or trying the opposite direction (products -> reactants) might help.

How to get internal minima?

After the calculation is finished, you shall load the mcd.tbl file into Excel or like and plot the Energy vs reaction coordinate. You shall check for hills and valleys. Some hills are TS-es and some valleys are internal minima. The others are just numerical actifacts. The number of artifacts can be greatly decreased by setting smaller stepsizes.

You shall start a local minimalization from every valley-bottom geometries which you think can belong to a minimum. If it is, you will get an optimized minumum. If it is an artifact you will find a minimum which you already have.

After that you shall find the TS candidates. There is exactly one TS between two minima, so the easiest way to find it is picking the highest peak between the two. Don't forget that TS calculations need a good Hessian and unlike for optimizations the automatically generated approximate is not good enough. You shall specify calcfc in the input to order analytically calculated Hessian. If that calculation would be too expensive you shall start a job with maxcycle=1, calcfc on a smaller level of calculation (HF instead of DFT, smaller basis set) and read it's Hessian by readfc in the real TS calculation.

1. TS₁ and it's initial guess. They are very close to each other.

2. Internal minimum and it's initial guess. They are very close to each other.

3. TS₂

4. The highest energy point between the internal minimum and the endpoint, therefore the initial guess of TS₂. Though it is pretty far from it, it is still a good initial guess since TS₂ was found by the default saddle point search method of the Gaussian 03 started from this geometry.

5. Artifact minimum. An optimization from here goes to the optimized endpoint.

6. Artificial TS. Since the minimum next to it found to be fake, we do not care about it.

Restarting a job

The job can be restarted at any step by deleting the .gjf and .out/log files belonging to the unfinsihed step and copying the chksavstepnum_of_the_last_completed_step.chk over the working checkpoint file. When you run mcd.exe again, it will not run the Gaussian for the steps which have an existing .out/log file on the disk.

Input file reference

The program reads the parameters you set from the mcd.par file. This file is a human-readable text file containing sections. Each section has a header and some lines below it. You can download an input file and use as template.

[checkpoint] section

This section contains only one line with the name of the working checkpoint file. This must have the optimized reactant geometry and should have a good Hessian matrix.

[job] section

This section will be copied to the Gaussian input file as header and route section. The opt and geom keywords should not be included, this is given by the program.

[scan] section

This section defines the active coordinates.

• B 8 13 3.4 1.5
  This definition line means that a bond streching coordinate between atoms 8 and 13 is created. It's value in the reactants is 3.4A, in the products it is 1.5A.

• A 1 2 3 120 86
  This line defines an angle bend coordinate between atoms 1 2 3 which. The starting value is 120, the ending is 86 degrees.

• D 1 2 3 4 120 + 150
  This definition defines a torsion coordinate between atoms 1, 2, 3, 4. The starting value is 120, the ending is 150 degrees, the direction is positive.

[frozen] section

This section defines frozen coordinates like
B 1 2 1.1
A 1 2 3 30
D 1 2 3 4 86
This section can be skipped if no frozen coordinates exist.

[mcd] section

This section contains seven lines. Each contains a keyword and a number. All keywords are neccesary and their order is fixed. Below you can see them, their meaning and their suggested value:

• blocksize: the number of SCF+grad+geomopt steps the Gaussian makes before the result is examined. Too high value increase computational time. Too low has the chance of bad move if the convergence limits are loose enough. Suggested value: 3

• stepsize: maximum size of the change of one coordinate in Angstroms (sum of the active coordinate changes/step). The actual stepsize depends on the convergence of the previous step. Too high value has the risk of error, too low wastes computational time. Suggested value: 0.2×number of active coordinates

• maxstep: maximum active coordinate change in a step in Angstroms. Too high has big risk of error, too low wastes computational time Suggested value: 0.2.

• rmsstepmin: the actual trust radius is stepsize×(rms_displacement-rmsstepmin)/(rmsstepmax-rmsstepmin). By setting rmsstepmin and max you can set the convergence level of the geometry optimization in the steps. Setting it too high can increase the number of artificial minima, setting it too low increase computational cost. Suggested value: 0.001

• rmsstepmax if the RMS Displacement of the geometry optimization is bigger then this, the step is not accepted and another blocksize geometry optimization cycles are forced on the step (restart). Suggested value: 0.05

• forcelimit if the Maximum Force of the geometry optimization is bigger then this, a restart is forced. Suggested value: 0.01

• steplimit if the Maximum Displacement of of the geometry optimization is bigger then this, a restart is forced. Suggested value: 0.3

[opt] section

This section contains optimizer parameters which are copied to the Gaussian input. loose,tight,gdiis,calcfc and such parameters can be used. If you insert readfc, then the Hessian of the previous point is used creating the next. It is encouraged at small stepsized but not for large ones.

Error messages

The program sometimes ends with an error message. The meaning and possible fixes are listed here.

• Cannot find final structure: The final step did not provided a complete gaussian output. Check stepend.out and stepend2.out for reasons. If you see curvilinear step not converged error message in the output, you set too high convergence limit in the [scan] section

• "opt" or "geom" cannot be in the input job section: Error in input file. opt and geom keywords cannot be in the [job] section, you can set your parameters for the opt keyword in the [opt] section.

• '[mcd] field is needed and must have all 7 values': the input must contain the [mcd] section and the section must have all variables set. Use the example input as template.

• the order of values in [mcd] field is fixed...: the [mcd] section must hold the variables in the same order as the example input .

• No scan coordinate section found in the input: the [scan] section holds the list of the active coordinates therefore must exist

• 'Too many words in scan/frozen line' or Wrong number of words in...: Type mismatch in the [scan] section.

• Unknown type in scan/frozen line: The active coordinates can be bondstrechings, anglebendings and torsions

• (Non)Optimized parameters section not found in filename: The gaussian job on filename was unsuccesful. Check it for reasons.

• End of file while reading Hessian and Cannot read complete second derivative matrix: Uncomplete Hessian matrix was found in the last Gaussian file. Check it for reasons.

Warning messages

Some errors are not so fatal that makes termination of the calculation inevetable. Than a warning message is written to the screen and to the mcd.out. You shall consider if they are ignorable or they are signs of a problem.

• the input structure is not perfectly optimized or the scan/frozen coordinates are not completely accurate: The step_1st.gjf orders Gaussian to optimize the initial structure found in the working checkpoint file applying the initial values of the active coordinates from the input file. Since the checkpoint is supposed to contain the optimized reactant geometries and the initial values of the active coordinates supposed to be equal to their values in the reactants, this step should converge in one geometry optimization cycle. If it does not, than you get this warning. Reasons can be incomplete or loose optimization of the reactants, inaccurate (like 2.15 instead of 2.14878) coordinate value or error in the input.

• Warning: unexpected change in coordinate N: it means that this active or frozen coordinate has a different value in the output of the geometry optimization step than it was defined in the input. It is the problem of the redundant coordinate system modifier of the Gaussian, since the coordinates are correlated, changing one changes the other as well. Unless your active and frozen coordinates are dependent, fixing their value is possible, but Gaussian sometimes fails to make it. Every time you see this warning you shall look at the output to decide if the structure is acceptable or not. Usually the difference is small and diminishes in the next step. If it does not or the error is inacceptable, the only solution can be decreasing the stepsize.

Support

I tried to make as clear and readable code as possible but I know that clear and readable code is just as rare as honest politican. Feel free to contact me:

• you need help in modifying the program, creating derivative work

• you need help in recode it in a different language

• the program behaves differently than written in this manual (bug)

• the manual is unclear

• you get a crash like:
  Runtime error 207 at 0x00401062
  0x00401062
  0x004010D5
  If you report a crash, please send me the input files as well so I can examine the reason of the crash

• If you have an idea to make it better, need a new functionality

Do not ask for help in selecting appropriate active coordinate for your reaction. This program is not a black box machine for dummies but a tool for chemists. The active coordinate can only be defined on the basis of your mechanism theories. If you don't have theories, too bad. :-)

You can contact me by sending a mail to imre.berente@mailbox.hu.
I hope this program will become a useful tool for you!
Imre Berente, 2006. Apr. 28, Székesfehérvár, Hungary

The advantage of unrestricted calculations is that they can be performed very efficiently. The disadvantage is that the wave function is no longer an eigenfunction of the total spin, <S²>, thus some error may be introduced into the calculation. This error is called spin contamination.
How does spin contamination affect results
Spin contamination results in having wave functions which appear to be the desired spin state, but have a bit of some other spin state mixed in. This occasionally results in slightly lowering the computed total energy due to having more variational freedom. More often the result is to slightly raise the total energy since a higher energy state is being mixed in. However, this change is an artifact of an incorrect wave function. Since this is not a systematic error, the difference in energy between states will be adversely affected. A high spin contamination can affect the geometry and population analysis and significantly affect the spin density. As a check for the presence of spin contamination, most ab initio programs will print out the expectation value of the total spin, <S²>. If there is no spin contamination this should equal s(s+1) where s equals 1/2 times the number of unpaired electrons. One rule of thumb which was derived from experience with organic molecule calculations is that the spin contamination is negligible if the value of <S²> differs from s(s+1) by less than 10%. Although this provides a quick test, it is always advisable to double check the results against experimental evidence or more rigorous calculations.

Spin contamination is often seen in unrestricted Hartree-Fock (UHF) calculations and unrestricted Møller-Plesset (UMP2, UMP3, UMP4) calculations. It is less common to find any significant spin contamination in DFT calculations, even when unrestricted Kohn-Sham orbitals are being used.

Unrestricted calculations often incorporate a spin annihilation step which removes a large percentage of the spin contamination from the wave function at some point in the calculation. This helps minimize spin contamination but does not completely prevent it. The final value of <S²> is always the best check on the amount of spin contamination present. In Gaussian, the option "iop(5/14=2)" tells the program to use the annihilated wave function to produce the population analysis. I am not aware of any programs that use the annihilated wave function to perform the geometry optimization.

Restricted open shell calculations
It is possible to run spin-restricted open shell calculations (ROHF). The advantage of this is that there is no spin contamination. The disadvantage is that there is an additional cost in the form of CPU time required in order to correctly handle both singly occupied and doubly occupied orbitals and the interaction between them. As a result of the mathematical method used, ROHF calculations give good total energies and wave functions but the singly occupied orbital energies don't rigorously obey Koopman's theorem. When it has been shown that the errors introduced by spin contamination are unacceptable, restricted open shell calculations are the best way to get a reliable wave function.

Within the Gaussian program, restricted open shell calculations can be performed for Hartree-Fock, density functional theory, MP2 and some semiempirical wave functions. The ROMP2 method does not yet support analytic gradients, thus the fastest way to run the calculation is as a single point energy calculation with a geometry from another method. If a geometry optimization must be done at this level of theory, a non-gradient based method such as the Fletcher-Powell optimization must be used (note that the G94 manual implies that this may not still be functional for all cases).

Spin projection methods
Another approach is to run an unrestricted calculation then project out the spin contamination after the wave function has been obtained (PUHF, PMP2). A spin projected result does not give the energy obtained by using a restricted open shell calculation. This is because the unrestricted orbitals were optimized to describe the contaminated state rather than being optimized to describe the spin projected state.

A similar effect is obtained by using the Spin Constrained UHF method (SUHF). In this method the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice small positive values remove most of the spin contamination.

Half-electron approximation
Semiempirical programs often use the half electron approximation for radical calculations. The half electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in a consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single determinant calculation is used, there is no spin contamination. The consistent total energy makes it possible to compute singlet-triplet gaps using RHF for the singlet and the half electron calculation for the triplet. Koopman's theorem is not obeyed for half electron calculations. Also, no spin densities can be obtained. The Mulliken population analysis is usually fairly reasonable.

Further information
Some discussion and results are in
W. J. Hehre, L. Radom, P. v.R. Schleyer, J. A. Pople "Ab Initio Molecular Orbital Theory" Wiley (1986) An article that compares unrestricted, restricted and projected results is
M. W. Wong, L. Radom J. Phys. Chem. 99, 8582 (1995)

Some specific examples and a discussion of the half electron method are given in
T. Clark "A Handbook of Computational Chemistry" Wiley (1985)

A more mathematical treatment can be found in the paper
J. S. Andrews, D. Jayatilake, R. G. A. Bone, N. C. Handy, R. D. Amos Chem. Phys. Lett. 183, 423 (1991)

SUHF results are examined in
P. K. Nandi, T. Kar, A. B. Sannigrahi Journal of Molecular Structure (Theochem) 362, 69 (1996)

An expanded version of this article will be published in "Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems" by David Young, which will be available from John Wiley & Sons in the spring of 2001.

1. Music - Multipurpose Simulation Code.

This is a code written in Fortran by Snurr Research Group in Northwestern University. This code is design to do a object oriented molecular simulation. Currently, it is used for diffusion and adsorption in zeolite but it can be extended since it is a basic code. It is available free in http://zeolites.cqe.northwestern.edu/Music/music.html.

2. Towhee or MCCCS Towhee.

This software is written by J. Ilja Siepmann's research group in 1994. The simulation is molecular based on Monte Carlo. MCCCS itself is abbreviation of Monte Carlo for Complex Chemical System. You can browse the capability and also download it in http://towhee.sourceforge.net/index.html.

3. DL_Poly

DL_POLY is a general purpose serial and parallel molecular dynamics simulation package developed at Daresbury Laboratory by W. Smith, T.R. Forester and I.T. Todorov. DL_Poly is licensed software but you can have it individually under an academic license, means free for scientist for not commercial uses. More information can be seen in http://www.cse.scitech.ac.uk/ccg/software/DL_POLY/index.shtml

Ria Armunanto¹, Christian F. Schwenk, A. H. Bambang Setiaji¹ and Bernd M. Rode^, Department of Theoretical Chemistry, Institute of General, Inorganic and Theoretical Chemistry, University of Innsbruck, Innrain 52a, A-6020, Innsbruck, AustriaReceived 20 June 2003;  accepted 20 August 2003. ; Available online 22 September 2003.

Abstract

Classical and quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) simulations have been performed to describe structural and dynamical properties of Co²⁺ in water. The most important region, the first hydration shell, was treated by ab initio quantum mechanics at unrestricted Hartree–Fock (UHF) level using the LANL2DZ ECP basis set for Co²⁺ and the double-ζ plus polarization basis set for water. For the rest of the system newly constructed three-body corrected potential functions were used. A well-structured rigid octahedron was observed for the stable first hydration shell showing no first shell water exchange process within a simulation time of 11.9 ps. For second hydration shell ligands, a mean residence time of 28 ps was observed. Librational and vibrational motions as well as the ion–oxygen motion were investigated by means of velocity autocorrelation functions showing significant differences between classical and QM/MM results.

 

Article Outline

1. Introduction

2. Methodology

2.1. Construction of potential functions

2.2. Simulation performance

2.3. QM/MM molecular dynamics simulation

2.4. Velocity autocorrelation functions

2.5. Mean residence times and reorientational times

3. Results and discussion

3.1. Structural data

3.2. Dynamical data

3.2.1. Librational and vibrational motions

3.2.2. Ligand exchange processes

4. Conclusion

Acknowledgements

References

1. Introduction

The hydration structure of transition metal ions is of much interest, since they have several key functions in biomolecular systems [1]. The methods used for structural investigations of hydrated metal ions can be classified into three types: scattering methods such as X-ray diffraction (XD) and neutron diffraction (ND), spectroscopic methods such as extended X-ray absorption fine structure (EXAFS) and nuclear magnetic resonance (NMR), and the tools of theoretical chemistry, including a wide variety of different simulation techniques [2]. Simulation methods such as Monte Carlo (MC), classical molecular dynamics (MD) and hybrid quantum mechanical/molecular mechanical (QM/MM) simulations, have proven to be a strong alternative to experiments in particular for investigations where experiments reach their limitations [3, 4, 5, 6, 7, 8 and 9]. Classical molecular dynamics simulation methods have been widely used, inter alia to analyze solutions of alkali metal ions and transition metal ions in water or ammonia [5, 10, 11, 12, 13, 14, 15, 16 and 17], and it has been shown in many cases, that pair potentials are inadequate for such systems [18, 19, 20, 21 and 22]. Non-additive terms (3,4,5,…,n-body) thus play an important role and should, therefore, be included in the potential functions. However, this procedure is usually restricted to three-body terms, as the construction of higher energy surfaces becomes very complicated.

Full ab initio quantum mechanical treatment could include all n-body terms, but is still far beyond current computer capacities. To reduce the time demand without loosing accuracy of the results the system can be partitioned, however, into the region of the ion with its first hydration shell, which is treated quantum mechanically, and the classically described remaining region which uses three-body corrected ab initio evaluated analytical potential functions. This method is referred to as hybrid QM/MM simulation technique.

Ligand exchange rates and residence times of ligands in the coordination shell of ions are important dynamical parameters to understand the reactivity of these ions in chemical and biological systems. The exact structure and in particular the dynamics of hydrated transition metal ions are highly sensitive to the accuracy of simulation techniques, and it has been shown in several cases [7, 8, 23, 24, 25 and 26] that only ab initio QM/MM simulations reach a sufficient level of accuracy. In the present work, we have extended these investigation to hydrated Co²⁺, in order to obtain structural and dynamical properties of this ion, which plays a quite significant role in solution chemistry and biomolecules.

2. Methodology

2.1. Construction of potential functions

New potential functions for Co²⁺–H₂O and H₂O–Co²⁺–H₂O interactions were constructed from ab initio quantum mechanical calculations at unrestricted Hartree–Fock (UHF) level using the double-ζ plus polarization basis set for water and the LANL2DZ ECP basis set for Co²⁺ [27 and 28]. These energies were fitted to analytical functions using the Levenberg algorithm. Experimental gas phase values were used for the water geometry (O–H=0.9601 Å, H–O–H=104.47°) and kept constant throughout the energy calculations [29]. Oxygen and hydrogen charges were set to −0.6598 and 0.3299, respectively, in agreement with the BJH-CF2 water model [30, 31 and 32] used for water–water interactions in this work. The basis set superposition error (BSSE) [33] in this system is very small amounting 0.183 kcal mol⁻¹. About 2719 ab initio energies points were calculated using the Turbomole program [34, 35 and 36]. The minimum energy for the Co²⁺–H₂O interaction was found to be −86.5 kcal mol⁻¹ at a distance of 2.02 Å. An MP2 calculation yielded nearly the same Co–O distance (2.00 Å) and a slightly lower energy (2.87 kcal mol⁻¹) showing small electron correlation effects in this system. According to previous results [24 and 37] the usage of UHF calculations with DZP basis set seems a reasonable compromise between accuracy and computational effort, also minimizing possible BSSE errors [37]. The limitation of the method could be seen rather in the size of the QM region and not so much in the QM level of calculation.

The ab initio calculated Co²⁺–H₂O energies were fitted to an analytical function of the following form:
 

(1)

where M denotes Co²⁺ and i water atoms; A, B, C and D are the optimized parameters summarized in Table 1, and q represents the atomic charges.

Table 1. Optimized parameters of the analytical Co²⁺–H₂O pair potential function

A total of 13,631 ab initio energy points were generated to describe the H₂O–Co₂₊–H₂O energy surface and to construct a three-body correction function
 

where ab and 2bd denote ab initio and two-body energies; MW and WW indicate ion–water and water–water interactions; r₁, r₂ and r₃ correspond to ion–water(1), ion–water(2) and water(1)–water(2) distances, respectively. The obtained three-body correction function is

where CL, set to 6.0 Å, is the cut-off limit beyond which three-body terms are negligible.

2.2. Simulation performance

The simulations were performed for one Co²⁺ and 499 water molecules in an elementary cubic box of 24.6 Å side length, at 298.16 K, which corresponds to a density of 0.99072 g cm⁻³. Periodic boundary conditions were applied to the simulation box and the temperature was kept constant by the Berendsen algorithm [38 and 39]. The flexible BJH-CF2 water model which includes an intramolecular term was used [30, 31 and 32]. Accordingly, the time step of the simulation was set to 0.2 fs, which allows for explicit movement of hydrogens. A cut-off of 12.0 Å was set except for O–H and H–H non-Coulombic interactions for which it was set to 5.0 and 3.0 Å . The reaction field method was used to account for long-range electrostatic interactions [40].

2.3. QM/MM molecular dynamics simulation

A classical molecular dynamics simulation was carried out for 60.0 ps after 60.0 ps of equilibration using the pair plus three-body function. Subsequently, the QM/MM simulation was performed for 11.9 ps after 4 ps of re-equilibration. The ab initio quantum mechanical treatment was applied to the ion and the full first hydration shell, and for the remaining MM region the same 2 + 3-body potential as in the classical simulation was used. According to the Co–O RDF of the classical simulation, the QM radius was set to 3.8 Å to fully include the first hydration shell. A smoothing function was applied to the transition region between the QM and the MM regions [38]. The force of the system, F_system, is defined as
 

where F_MM is the MM force of the full system, F_QM the QM force in the QM region, F_QM/MM the MM force in the QM region. S denotes the smoothing function [41]. The use of this smoothing function and the algorithm of our QM/MM simulation allows the water ligands to migrate freely between the two regions with a steady transition of forces. In this context, the flexibility of the MM water molecules is another important factor, as this flexibility is thus given for ligands both inside and outside the QM region.

2.4. Velocity autocorrelation functions

The evaluation of spectral properties such as librational and vibrational frequencies of water molecule motions was carried out using velocity autocorrelation functions (VACFs), C(t), defined as
 

(5)

where N is the number of particles, N_t is the number of time origins t_i, and denotes a certain velocity component of particle j. The power spectrum of the VACF was calculated by Fourier transformation. A correlation length of 2.0 ps was used to obtain the power spectra with 4000 (classical) and 2000 (QM/MM) averaged time origins. Librational and vibrational frequencies of water molecules were computed using the approximative normal coordinate analysis [42]. Six scalar quantities Q₁, Q₂, Q₃, R_x, R_y, R_z define the symmetric stretching, bending and asymmetric stretching vibrations, and rotations around the three principal axes of the water molecules.

2.5. Mean residence times and reorientational times

The mean residence time (MRT) of water molecules in the second hydration shell of Co²⁺ was calculated with the following formalism proposed by Impey et al. [43]:
 

(6)

where n_ion(t) is the number of water molecules which lie initially within the coordination shell and are still there after a time t elapsed. The parameter t^* is introduced to avoid counting of water molecules leaving the coordination shell only temporarily and returning to it within t^*. The parameter of t^* was set to 2.0 ps in accordance with Impey [43].Reorientational time correlation functions (RTCFs) of water molecules were calculated as
 

(7)

where P_l is the Legendre polynomial of lth order and is a unit vector along the three principal axes i defined in a fixed coordinate frame as the rotations above.As exponential decay is assumed for the MRTs and RTCFs, an exponential fit was used
 

where a and τ are the fitting parameters, and τ describes the corresponding relaxation time.

3. Results and discussion

3.1. Structural data

The radial distribution functions (RDFs) of Co²⁺–O and Co²⁺–H together with their integration numbers obtained from the classical and the QM/MM simulations are displayed in Fig. 1. Two well-defined peaks are obtained from both simulations indicating first and second hydration shell. The first QM/MM peak is shifted closer to the ion in comparison with the classical peak, reflecting a remarkable influence of higher n-body effects. The sharpness of the first peak corresponds to a highly structured, rather rigid first hydration shell. The zero-value Co–O RDF between the two peaks indicates that no exchange process occurred within the simulation time. The broad second peaks observed in both classical and QM/MM simulations shows a high flexibility of water molecules in this shell. The first peak obtained from the QM/MM simulation is centered at 2.17 Å, while the classical simulation shifts it to 2.27 Å. The second shell peaks are centered around 4.6 Å in both simulations, but with a broad plateau in the classical case. These results are in good agreement with Co–O distances of the first and second hydration shell obtained by EXAFS, XD and ND experiments [2 and 44]. The average Co–O distance obtained from the QM/MM simulation (2.17 Å) is only slightly higher than XD (2.09 Å) and EXAFS (2.08 Å) data, the difference being probably due to concentration effects [2].

Fig. 1. Co–O and Co–H radial distribution functions and their corresponding integration numbers obtained from QM/MM (solid line) and classical (dotted line) MD simulations.

Coordination number distributions of hydrated Co²⁺ obtained from the classical and the QM/MM simulation are displayed in Fig. 2. The obtained six-coordinated complex in the first hydration shell (100% occurrence) is in agreement with EXAFS data [2 and 44], whereas the classical simulation gives a slightly lower value (5.9). Classical pair plus three-body simulations often allow a correct description of rough structural data as first shell coordination numbers. However, the too repulsive three-body potential apparently causes a small shift of the first hydration shell to a larger distance. This rather small difference between classical and QM/MM result in the first coordination shell then induces larger deviations in the second shell caused by different ligand orientations in the first shell and polarization effects of first shell ligands not accounted for by the classical potentials. The classical simulation thus strongly overestimates the second shell coordination number yielding a value of 22.7 whereas the QM/MM value of 15.9 is closed to the value of 14.8 estimated from XD [2]. The classical simulation thus also yields a broader coordination number distribution (18–28), while the QM/MM simulation gives values between 11 and 19.

Fig. 2. Coordination number distributions of Co²⁺ in water obtained from (a) QM/MM and (b) classical MD simulations.

The angular distribution function (ADF) of O–Co²⁺–O angles is shown in Fig. 3. The ADF obtained from the QM/MM simulation displays two peaks located at 90° and 180°. The first peak located at 91° is caused by two neighboring oxygens, and is in good agreement with the angle deduced from mass spectroscopic analysis (90°) [2 and 44]. The second peak culminates at 173° indicating an octahedral arrangement of the water molecules in the first shell of Co²⁺ (Fig. 4), in agreement with XD data (see Table 3). The small artificial peak at 70° in the classical simulation is caused by short-lived sevenfold coordinated intermediates which have not been obtained in the QM/MM simulation.

Fig. 3. Angular Distribution Function of O–Co²⁺–O angles observed from classical (dotted line) and QM/MM (solid line) MD simulations.

Fig. 4. Octahedral structure of the first hydration shell of Co²⁺ in water.

Table 3. Structural parameters of the first hydration shell of Co²⁺ in water

Two further angles were defined to describe the orientation of the water molecules relative to the ion (θ and tilt). The θ angle is the angle between the O–Co vector and the water plane, while the tilt angle is defined as the angle between the O–Co vector and dipole vector of the water molecule. The QM/MM simulation yields a θ value of 171° and a tilt value of 8° (see Table 2), whereas the classical simulation shows slightly lower θ angles and nearly no tilt.

Table 2. Hydration parameters for Co²⁺ in aqueous solution obtained from QM/MM and classical MD simulations

The hydration energy values of −550 and −547 kcal mol⁻¹ obtained from classical and QM/MM simulation are considerably lower than the experimentally estimated value of −487 kcal mol⁻¹ [45]. This difference is probably caused by the specific assumptions necessary to assign single-ion values to thermocalorimetric measurements of salts [45].

3.2. Dynamical data

3.2.1. Librational and vibrational motions

The power spectra of VACFs for the librational motions R_x, R_y, R_z and the vibrational motions Q₁, Q₂, Q₃ obtained from the classical and the QM/MM simulations are displayed in Fig. 5 and their frequencies are summarized in Table 4. The order of R_z<R_x<R_y is found in the second hydration shell and the bulk for QM/MM as well as classical simulation. In the first shell the QM/MM order is different (R_z<R_y<R_x), as already observed in previous simulations [7 and 23]. The R_z value in the first hydration shell (QM/MM simulation) is only slightly red-shifted since the rotation around the dipole axis is not energetically restricted. In contrast, the R_x QM/MM frequency is strongly blue-shifted due to the ligand fixation by the ion, as in the case of Ni²⁺ [7], Ca²⁺ [23], Fe²⁺ and Fe³⁺ [52]. The R_y value is nearly unchanged. The order R_x>R_y obtained from the QM/MM simulation contradicts the classical result, revealing some inadequacy of the 2 + 3-body function. In the second hydration shell, the librational motions of the classical and the QM/MM simulation are rather similar, showing slightly higher values in the QM/MM case. The frequencies of the bulk phase are in good agreement with the values obtained for pure liquid water (BJH model) as listed in Table 4.

Fig. 5. Power spectra of rotational modes R_z, R_x, R_y and vibrational modes Q₂, Q₁, Q₃ for water molecules in the first hydration shell obtained from QM/MM (solid line) and classical (dotted line) MD simulations.

Table 4. Librational and vibrational frequencies of water molecules in the first and second hydration shell of Co²⁺ in water and bulk

In comparison with the bulk, the stretching frequencies Q₁ and Q₃ of the first hydration shell from the QM/MM simulation are blue-shifted, whereas the bending frequency Q₂ is red-shifted in accordance with previous simulations of other hydrated ions [7, 23, 24 and 52]. In contrast, the classical simulation fails to reproduce these effects, giving red-shifted stretching modes Q₁ and Q₃, and a nearly unchanged bending mode Q₂. The frequency difference () between Q₁ and Q₃ is 76 cm⁻¹ in the QM/MM case, similar to Ni²⁺ [7] and V²⁺ [24] shifts.

The RTCF values of first (τ₁) and second (τ₂) order in the dipole moment direction of a water molecule are summarized in Table 5. The correlation function for l=1 is related to infrared line shapes and l=2 to Raman line shapes and NMR relaxation time [2]. The QM/MM results exhibit significantly larger relaxation times in comparison with classical data stressing once more the important role of many-body effects. Strongly increased relaxation times were only observed for the first hydration shell, outside of which the ion’s influence is rather weak. The bulk water relaxation times obtained from both simulations are in good agreement with previous simulations [24 and 51] and experimental data [2].

Table 5. Reorientational times of first and second order of water molecules in the first hydration shell of Co²⁺ in water

The power spectra of the Co²⁺–O stretching motion obtained from classical and QM/MM simulations are displayed in Fig. 6. The stretching frequency from the QM/MM and the classical simulations are 305 and 290 cm⁻¹, respectively. The corresponding QM/MM force constant of 69 N m⁻¹ is similar to the ones obtained from previous QM/MM MD simulations of Mn²⁺ and V²⁺ [24] with values of 59 and 70 N m⁻¹. The peak forms of the stretching vibration reflect some further inappropriate description of the dynamics in the first shell by the classical treatment.

Fig. 6. Power spectra of Co²⁺–oxygen vibrational modes obtained from QM/MM MD (solid line) and classical (dotted line) simulations.

3.2.2. Ligand exchange processes

The mean residence time of water molecules (τ) in the second hydration shell has been calculated from the QM/MM simulation with t^*=2.0 ps according to Impey [43]. A mean residence time of 28 ps in the second hydration shell was found, corresponding to the seven observed exchange processes during the 12 ps simulation. No experimental data for residence times of water ligands in the second hydration shell of Co²⁺ are available. In the first hydration shell no water exchange process was observed as a simulation time of 12 ps is by far too short compared to the experimental residence time of 10⁻⁷ s [2 and 53].

4. Conclusion

The inclusion of many-body effects appears mandatory in order to describe the hydration structure of Co²⁺ and its dynamical properties properly. The structural results clearly demonstrate that although rough structural properties as first shell coordination numbers are reproduced correctly, accurate ligand orientations are only available after including many-body terms through a quantum mechanical treatment. Therefore, the evaluation of the much more sensitive spectroscopic and dynamical data also requires this level of accuracy, and the inclusion of a larger number of water ligands in the surroundings of the ion appears desirable in further simulations.

 

Acknowledgements

Financial support for this work by the Austrian Science Foundation (FWF) (project P16221-N08) and a scholarship of the Austrian Federal Ministry for Foreign Affairs for R.A. are gratefully acknowledged.

 

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