Chemists are familiarized with the description of a chemical reaction as a continuous motion on certain path of the potential energy hypersurfaces connecting reactants with products. Those are considered minima in the hypersurface while an intermediate state known as the transition state would be a saddle point of higher energy. The height of the energy barrier separating reactants from products relates to the overall rate of reaction, the positions of the minima along the reaction coordinate give the equilibrium geometries of the species, and the relative energies relate to the thermodynamics of the process. All this is known as transition state theory.
The process to study a chemical reaction starts by obtaining proper geometries for reactants and products, follows by finding the position of the transition state, and finishes by computing as accurately as possible the relative energies relating the position of the species. To perform geometry optimizations searching for true minima in the potential energy surfaces (PES) is by now a well-established procedure (see section 4.2). An stationary point in the PES is characterized by having all the first derivatives of the energy with respect to each one of the independent coordinates equal to zero and the second derivatives larger than zero. First-order saddle points, on the contrary, have their second derivatives lower than zero for one coordinate, that is, they are maxima along this coordinate. A transition state is defined as a saddle point having only one negative second derivative along the specific coordinate known as the reaction coordinate. To simplify the treatment a special set of coordinates known as normal coordinates is defined in a way that the matrix of second derivatives is diagonal. A transition state will have one negative value in the diagonal of such a matrix.
In the present section we are going to characterize the rearrangement of dimetylcarbene to propene as an example of the use of the SLAPAF module in the calculation of both geometry minima and transition states. The geometries will be obtained at the CASSCF level of calculation. This is not an attempt to compute them accurately but a platform to show the use of the MOLCAS codes. At the obtained geometries different methods such as MP2, MRCI, ACPF, CASPT2, CCSD, and CCSD(T) will be used to compute relative energies. An accurate study of this reaction path has been recently performed [50] and we remit the reader to the proper reference for more details. Figure 4.7 describes the studied process.
In section 4.2 the detailed description of the
optimization procedures for stationary states was already
performed. We will not stress much this point here.
For the dimethylcarbene and propene we knew previously that
their equilibrium geometries have C
and C
symmetry
[50,51], respectively.
This knowledge facilitates much our work
because it allows to restrict the symmetry both in the
geometry optimization process and in the subsequent calculations.
For the dimethylcarbene we use the following input for the
STRUCTURE program:
&STRUCTURE &END Method casscf Restart /temp/$LOGNAME End of input &SEWARD &END Title Dimethylcarbene singlet C2-sym Symmetry XY Basis set C.ANO-S...3s2p1d. C1 0.000000 0.000000 0.683627 Angstroms C2 0.000000 1.214478 -0.162556 Angstroms End of basis Basis set H.ANO-S...2s1p. H1 -0.321236 1.116582 -1.206817 Angstroms H2 -0.431111 2.093839 0.311843 Angstroms H3 1.091662 1.390272 -0.180572 Angstroms End of basis PkThrs 1.0E-10 End of input &SCF &END Title Dmc Occupied 7 5 End of input &RASSCF &END Title Dmc Symmetry 1 Spin 1 Nactel 2 0 0 Inactive 6 5 Ras2 1 1 Iteration 50,25 LumOrb End of input &ALASKA &END End of input &SLAPAF &END Iterations 40 End of input
Observe that in the STRUCTURE namelist it is necessary to
include the keyword Method casscf or Method rasscf
to perform a geometry optimization based on the RASSCF input.
We have also used the option Restart /temp/$LOGNAME which
would allow us to restart the calculation from any of the programs
using the files stored in the directory /temp/$LOGNAME.
The calculations were performed restricting the geometry to the C
symmetry group. As will be explained below the active space for the
CASSCF calculation was selected to be two electrons in two orbitals
after testing larger spaces. The convergence process was not difficult
as it is shown by the SLAPAF output:
*****************************************************************************************
********** Energy Statistics for Geometry Optimization **********************************
*****************************************************************************************
Energy Grad Grad Step Estimated Hessian
Iter Energy Change Norm Max Element Max Element Final Energy Index
1 -117.01877510 .00000000 .020071 .010951 nrc005 -.162975 nrc011 -117.01987401 0
2 -117.02003731 -.00126221 .012173 .004582 nrc008 -.073781 nrc011 -117.02043010 0
3 -117.02041716 -.00037985 .003705 .002607 nrc007 -.030446 nrc011 -117.02045464 0
4 -117.02045993 -.00004277 .000764 .000437 nrc007 -.004111 nrc011 -117.02046129 0
5 -117.02046148 -.00000155 .000205 .000303 nrc005 .004293 nrc011 -117.02046185 0
6 -117.02046145 .00000003 .000289 -.000229 nrc011 -.002856 nrc011 -117.02046189 0
7 -117.02046172 -.00000028 .000160 -.000298 nrc005 -.001103 nrc005 -117.02046194 0
8 -117.02046185 -.00000012 .000235 .000248 nrc005 -.000565 nrc008 -117.02046185 0
Cartesian Displacement Gradient (Int. Coord.)
Value Threshold Converged? Value Threshold Converged?
+----------------------------------+----------------------------------+
RMS + .4983E-03 .1200E-02 Yes + .7096E-04 .3000E-03 Yes +
+----------------------------------+----------------------------------+
Max + .6347E-03 .1800E-02 Yes + .2479E-03 .4500E-03 Yes +
+----------------------------------+----------------------------------+
Geometry is converged
*****************************************************************************************
For the propene molecule the input file would be:
&STRUCTURE &END Method casscf Restart /temp/teolsa End of input &SEWARD &END Title Propene singlet Cs-sym Symmetry Z Basis set C.ANO-S...3s2p1d. C1 -2.4150580342 .2276105054 .0000000000 C2 .0418519070 .8733601069 .0000000000 C3 2.2070668305 -.9719171861 .0000000000 End of basis Basis set H.ANO-S...2s1p. H1 -3.0022907382 -1.7332097498 .0000000000 H2 -3.8884900111 1.6454331428 .0000000000 H3 .5407865292 2.8637419734 .0000000000 H4 1.5296107561 -2.9154199848 .0000000000 H5 3.3992878183 -.6985812202 1.6621549148 End of basis PkThrs 1.0E-10 End of input &SCF &END Title Propene Occupied 10 2 End of input &RASSCF &END Title Propene Symmetry 1 Spin 1 Nactel 2 0 0 Inactive 10 1 Ras2 0 2 Iteration 50,25 LumOrb End of input &ALASKA &END End of input &SLAPAF &END Iter 40 End of input
The convergence was simple for propene:
****************************************************************************************
********************************** Energy Statistics for Geometry Optimization *********
****************************************************************************************
Energy Grad Grad Step Estimated Hessian
Iter Energy Change Norm Max Element Max Element Final Energy Index
1 -117.12218420 .00000000 .010310 .004094 nrc010 .010109 nrc010 -117.12224754 0
2 -117.12225965 -.00007545 .002090 .000970 nrc010 .003538 nrc010 -117.12226368 0
3 -117.12226360 -.00000395 .000327 -.000262 nrc010 -.000845 nrc010 -117.12226382 0
Cartesian Displacement Gradient (Int. Coord.)
Value Threshold Converged? Value Threshold Converged?
+----------------------------------+----------------------------------+
RMS + .1034E-02 .1200E-02 Yes + .8731E-04 .3000E-03 Yes +
+----------------------------------+----------------------------------+
Max + .1787E-02 .1800E-02 Yes + .2621E-03 .4500E-03 Yes +
+----------------------------------+----------------------------------+
Geometry is converged
****************************************************************************************
****************************************************************************************
The obtained geometries will be described below.
The calculation of transition states remains as one of the most difficult problems in quantum chemistry [16]. There are many reasons for that. For instance, the mathematical procedures to find saddle points are not so well developed as those to find absolute minima. On the other hand low-level theoretical treatments do not help much in situations were the chemical bonds are partially or totally broken and the potential energy surfaces are extremely flat such as close to the transition state situation. Most important, the common experience on transition state structures is no so wide as in equilibrium structures and experimental techniques are not of much help in many cases.
One has to gain experience on the different strategies to compute transition states [52]. Although low-level methods such semiempirical, Hartree-Fock, and even many density functional procedures sometimes fail to produce reasonable transition state geometries they can be helpful on reducing the overall computation if they are used with caution. In systems in which there is no previous experience even for related situations it is better to obtain information from model systems and low-level methods before starting the true search. The best way to get an initial guess for a transition state geometry is probably to base the guess on the transition structure for a closely-related system obtained at the same level of calculation. If this is not available one has to proceed towards the same system with a lower level of theory. Apart from that, the method known as Linear Synchronous Transit [53], which represent an average situation between reactant and product geometries, is as effective as many other recipes.
The present case pretends to be an example of how to solve quantum
chemical problems using the tools supplied by MOLCAS. We are not
going to perform an exhaustive study of the problem. To start
with the transition state search we can take a new look to
Figure 4.7. From reactants to products the simplest
way to migrate an hydrogen atom from a terminal to the central
carbenoid center is to decrease the angle formed by the carbenoid
center (C
), the terminal carbon (C), and the hydrogen (H
),
elongate the C-H bond, and decrease the C-H
distance. The movement can be done keeping the hydrogen (H)
almost perpendicular to the three carbon planes and the spectator
methyl group as a rigid unit. This reaction path can be designed
to keep the C symmetry. We find in the literature a
study following this path [54] which we can use as
initial guess. Though it simplifies the calculations it is not,
however, advisable to use symmetry restrictions for geometry
optimizations when searching for transition states. As an initial
guess for the transition state structure we will take a
geometry where the hydrogen is at an intermediate distance from
C and C and almost perpendicular to the carbon plane,
although not completely. As we are not going to impose any
symmetry constraints is better not to start from a symmetric
situation.
In this case we have some clues about the location of the transition
state. In other cases we better take advantage of other programs
which perform automatic transition states search using for instance
the Linear Synchronous Transit approach or look for the Intrinsic
Reaction Coordinate. One possible strategy if one suspects which
is the reaction coordinate is to modify the geometry, fix the
internal coordinate (in the user-defined internal coordinate format)
related to the atom or atoms taking part in the reaction process,
and optimize the remaining coordinates. SLAPAF will print
in each iteration the gradient corresponding to the fixed coordinate.
In this case, for instance, we decrease the angle CCH,
defined as internal coordinate A4, fix the coordinate, and perform
an optimization of the remaining coordinates. We obtain:
Following internal coordinates are fixed A4 with a gradient of .528E-01 is frozen and the gradient is annihilated
If we follow the value of the gradient along the optimization path we will see how it decreases towards a point when it increases again. This point can be expected to be in the proximities of the searched saddle point. Sometimes it is difficult to proceed with these calculations. Other coordinates can be left fixed to help in the process; for instance here the coordinates related to the spectator methyl group.
For the present example we performed checking Hessian determinations
at the selected geometry using a semiempirical method such as AM1 and
the RHF method with small STO-3G
and 3-21G basis sets.
AM1 computed one large and one small negative
eigenvalues while RHF led to one single negative eigenvalue
(imaginary frequency 1400i cm
)
involving the H motion. This is a common value for imaginary
frequencies near the transition state structure when a bond is broken.
Some of these Hessian can be used as an input for SLAPAF.
We can anyway use the geometry to compute a numerical Hessian with
MOLCAS and obtain additionally the best suited coordinates to
perform the transition state search.
&STRUCTURE &END Method CASSCF Restart /temp/$LOGNAME End of input &SEWARD &END Title Transition state between Dimethylcarbene singlet and 1-Propene Basis set C.ANO-S...2s1p. C1 -0.082613 -0.623163 -0.107344 Angstrom C2 -1.206102 0.213341 -0.042268 Angstrom C3 1.210166 0.145247 0.015156 Angstrom End of basis Basis set H.ANO-S...1s. H1 -1.164741 1.308782 0.025435 Angstrom H2 1.121076 1.208918 0.274026 Angstrom H3 -2.201164 -0.207032 -0.169569 Angstrom H4 1.906873 -0.338974 0.698675 Angstrom H5 -0.864636 -0.466136 0.951618 Angstrom H6 1.673882 0.081890 -0.973450 Angstrom End of basis End of input &SCF &END Title Carbene-propene transition state Occupied 12 End of input &RASSCF &END Title Carbene-propene transition state Symmetry 1 Spin 1 Nactel 2 0 0 Inactive 11 Ras2 2 Iteration 50,25 LumOrb End of input &ALASKA &END End of input &SLAPAF &END Numerical Hessian TS Iter 0 End of input
The input files shown above contain the necessary tools to compute a
numerical Hessian in a minimal ANO-type basis set. No symmetry
restriction is imposed. We are using the CASSCF method to compute
the geometry of the transition state as well as the Hessian. The
selected active space has two electrons and two orbitals. We
previously tested other active spaces such as four electrons in
four orbitals which included also the carbenoid electrons on
C an a second correlating orbital, but all the tests suggested
that only two orbitals actually participate on the whole proccess.
The convenience of this active space is discussed later.
Observe that the SLAPAF input contains simultaneously
the keywords NUMErical and TS, with the number
of iterations equal to zero. If the Hessian is going to be used
in a following transition state search is important to compute it
in the presence of the keyword TS. If not SLAPAF
corrects automatically for negative or too small eigenvalues as
if the Hessian is going to be used in a search for a minimum.
The output of SLAPAF first displays a list of the curvature weighted non-redundant internal coordinates in which the relaxation has been performed. This is the default option. We will take the printed coordinates from the output and use them for the input of SLAPAF showed below.
In the output, the numerical force constant matrix follows together with the harmonic oscillator
analysis, which, strictly, is only valid at stationary points. An imaginary
frequency 1059.70i cm has been computed in the present example. The listing
of the eigenvalues and eigenvectors points out to the seventh mode as having the
largest negative eigenvalue. To analyze the eigenvalues is sometimes more important
than to analyze the frequencies of the harmonic analysis. If the harmonic analysis
is not performed in a true stationary point its solution is not unique. Therefore
is better to consider the number of negative eigenvalues of the force constant matrix.
Analyzing mode number seven we observe the largest
contribution of the coordinate q007 (see below) with strong participation of the
redundant coordinates b08 = Bond H5 C1 and a07 = Angle H5 C2 C1, those which
we could expect a participation in the reaction path of hydrogen H. The
second contribution in importance comes from the coordinate q002, which mainly
involves b06 = Bond H3 C2, an hydrogen which must be reoriented along the process.
In this case there is only one negative eigenvalue. If this Hessian is used in a transition state search, SLAPAF will follow a reaction path for the transition state relaxation using the mode with the largest negative eigenvalue unless otherwise specified by input. In case of not having any negative eigenvalue the lowest positive value (21st here) would be made negative.
Eigenvalues ----------- .51109327 .48036140 .48745710 .32415493 .51923276 .50532072 -.04276585 .19122738 .46231977 .14771409 .04938337 .13109600 .10434187 .05319556 .11799430 .18243787 .09007300 .08648291 .11639532 .10519348 .00242778 ... Store Original mode: 7 Reaction mode ------------- mat. size = 21x 1 .04366171 -.50761438 .12658584 .09581436 -.04791768 .04818743 .63448362 -.14076455 .07050982 .05549477 .03009767 -.03940238 -.15831088 -.19356166 .36612094 .12196561 -.23725910 -.01258946 .02595128 -.06533074 -.06974532
When computing the Hessian without the option TS and still wanting to use it as input in a transition state search it is important to take the first printed force constant matrix, not the second one, which would have been transformed for the case of a minima. This can be easily observed by the output message:
Some negative eigenvalues have been corrected
before the second force constant matrix.
We can use then the computed Hessian for the subsequent transition state search with SLAPAF. The transition state search must be performed in the same coordinate space as that used to compute the Hessian. Otherwise the Hessian is useless. In addition to the Hessian is strongly recommended to use the curvature weighted non-redundant internal coordinates defined by SLAPAF to perform the optimization. This type of coordinates have favorable properties, especially in optimizations of van der Waals complexes and transition states. The usual non-redundant coordinates lead sometimes to undesired degeneracies between coordinates which have different influence in the optimization process. The weighting introduced in the coordinates helps to resolve these problems, which are more severe in cases such a transition state search, when the geometry of the system can be extremely distorted. In case of not using the coordinates displayed by SLAPAF, the program will print a message in which it will point out on the redundant coordinates which must be included in the description of the non-redundant coordinates because they become important along the optimization path. Option RTHR in SLAPAF can be used for this purpose. The output message would be, for instance:
Decrease the theshold for redundant coords 27 6 19
SLAPAF can be also supplied with option MODE, which tells the program the index of the mode it should follow in the transition state search.
Now we can submit the calculation for the transition state search. Observe that, as in the previous Hessian calculation, the option Restart has been used in STRUCTURE. This time, however, the saving directory is $WorkDir because it is strongly recommended not to remove the scratch directory in this case. For a transition state search to follow the optimization path is even more important than for a minimum search. In this case one may want to look at each one of the output files generated by STRUCTURE. The output files of each complete iteration are stored in the $WorkDir directory under the names $RANDOM.save.$iter, for instance: 12345.save.1, 12345.save.2, etc. You should not remove the $WorkDir directory if you want to keep them. As the search can proceed for strange conformations could be advisable to use the keyword RTRN in the SEWARD program. SEWARD and SLAPAF have both a criterion for printing bonds, valence angles, and torsional angles. The criterion is based on the distance between atoms. The default is set to 3.5 au. This means that if three atoms are separated, two by two, by more than 3.5 au the angle between them will not be printed. This can be a problem in transition states where the geometries differ from the standard ones. With the option RTRN in SEWARD set to a higher value one obtains a more informative (though very large sometimes) SEWARD output. Normally one suspects from the beginning of a search which are the important atoms to follow and guess the maximum distance among them.
We will enlarge now the basis set size to C 
and H 
to perform the final transition state search:
&STRUCTURE &END
Method CASSCF
Restart $WorkDir
End of input
&SEWARD &END
Title
Transition state between
Dimethylcarbene singlet
and 1-Propene
Basis set
C.ANO-S...3s2p1d.
C1 -0.082613 -0.623163 -0.107344 Angstrom
C2 -1.206102 0.213341 -0.042268 Angstrom
C3 1.210166 0.145247 0.015156 Angstrom
End of basis
Basis set
H.ANO-S...2s1p.
H1 -1.164741 1.308782 0.025435 Angstrom
H2 1.121076 1.208918 0.274026 Angstrom
H3 -2.201164 -0.207032 -0.169569 Angstrom
H4 1.906873 -0.338974 0.698675 Angstrom
H5 -0.864636 -0.466136 0.951618 Angstrom
H6 1.673882 0.081890 -0.973450 Angstrom
End of basis
End of input
&SCF &END
Title
Carbene-propene transition state
Occupied
12
End of input
&RASSCF &END
Title
Carbene-propene transition state
Symmetry
1
Spin
1
Nactel
2 0 0
Inactive
11
Ras2
2
Iteration
50,25
LumOrb
End of input
&ALASKA &END
End of input
&SLAPAF &END
TS
Iter
10
Internal Coordinates
b01 = Bond C2 C1
b02 = Bond C3 C1
b03 = Bond C3 C2
b04 = Bond H1 C2
b05 = Bond H2 C3
b06 = Bond H3 C2
b07 = Bond H4 C3
b08 = Bond H5 C1
b09 = Bond H5 C2
b10 = Bond H6 C1
b11 = Bond H6 C3
a01 = Angle C3 C1 C2
a02 = Angle H5 C1 C2
a03 = Angle H5 C1 C3
a04 = Angle H1 C2 C1
a05 = Angle H3 C2 C1
a06 = Angle H3 C2 H1
a07 = Angle H5 C2 C1
a08 = Angle H5 C2 H1
a09 = Angle H5 C2 H3
a10 = Angle H2 C3 C1
a11 = Angle H4 C3 C1
a12 = Angle H4 C3 H2
a13 = Angle H6 C3 C1
a14 = Angle H6 C3 H2
a15 = Angle H6 C3 H4
a16 = Angle C2 H5 C1
t01 = Dihedral C1 H5 C2 H1
t02 = Dihedral C1 H5 C2 H3
t03 = Dihedral C2 H5 C1 C3
t04 = Dihedral H1 C2 C1 C3
t05 = Dihedral H1 C2 C1 H5
t06 = Dihedral H2 C3 C1 C2
t07 = Dihedral H2 C3 C1 H5
t08 = Dihedral H3 C2 C1 C3
t09 = Dihedral H3 C2 C1 H5
t10 = Dihedral H4 C3 C1 C2
t11 = Dihedral H4 C3 C1 H5
t12 = Dihedral H5 C2 C1 C3
t13 = Dihedral H6 C3 C1 C2
t14 = Dihedral H6 C3 C1 H5
Vary
q001 = .77806790 b01 + -.23263729 b02 + .03698923 b03 + -.33174975 b04 &
+ .02410283 b05 + -.23451362 b06 + .06212440 b07 + .15581124 b08 &
+ .22982154 b09 + -.01984946 b10 + .03151407 b11 + -.05689732 a01 &
+ -.02770294 a02 + -.00730798 a03 + -.11236506 a04 + -.12070454 a05 &
+ .20507710 a06 + -.06890971 a07 + -.00633941 a08 + -.02886562 a09 &
+ .07823155 a10 + -.01699450 a11 + -.02721481 a12 + -.01707788 a14 &
+ -.02692913 a15 + .05871016 a16 + -.01359793 t01 + .00834789 t02 &
+ .00627099 t03 + .00457080 t04 + .00184937 t05 + -.00230028 t06 &
+ -.00800006 t07 + -.00479586 t08 + -.00455485 t09 + -.00562839 t10 &
+ -.01003311 t11 + .00144096 t12 + .00292178 t13 + -.00499993 t14
q002 = -.35373146 b01 + .09848028 b02 + -.38096645 b04 + .18103993 b05 &
+ -.42249234 b06 + .24528444 b07 + -.40949317 b08 + .30630214 b09 &
+ .06172202 b10 + .24996936 b11 + .07513992 a01 + .16153018 a02 &
+ .02957385 a03 + -.03365152 a04 + -.08092817 a05 + .12561615 a06 &
+ -.14492292 a07 + -.05262863 a08 + -.03449850 a09 + -.04055291 a10 &
+ .08336985 a11 + -.05323380 a12 + .12108897 a13 + -.05281953 a14 &
+ -.06046413 a15 + -.02331772 a16 + -.00606644 t01 + .00242046 t02 &
+ -.01057108 t03 + .04973474 t04 + .02397867 t05 + -.02780766 t06 &
+ .01546831 t07 + -.01148321 t08 + -.01790470 t09 + -.02077297 t10 &
+ .02000678 t11 + .01128912 t12 + -.02874607 t13 + .01514408 t14
q003 = .19035952 b01 + .43117417 b02 + -.00361945 b03 + .36206189 b04 &
+ .51380872 b05 + .09339695 b06 + .35619922 b07 + .10822640 b08 &
+ .05042783 b09 + .05511067 b10 + .30803583 b11 + -.23187877 a01 &
+ -.01688201 a02 + -.12324802 a03 + .16292907 a04 + -.16005165 a05 &
+ -.00615955 a06 + .00934477 a07 + -.00993281 a08 + -.00926466 a09 &
+ .09027944 a10 + -.03792120 a11 + -.01267087 a12 + -.03771977 a13 &
+ -.01176009 a14 + .00582219 a16 + .01739020 t01 + .01533760 t02 &
+ .01372517 t03 + -.00822342 t04 + .01006055 t05 + .00602579 t06 &
+ -.01072596 t07 + -.00495049 t08 + .01254959 t09 + -.01430127 t11 &
+ -.01783521 t12 + .00849490 t13 + -.00938560 t14
q004 = -.01164175 b01 + -.41748995 b02 + .01082116 b03 + .51997324 b04 &
+ .31616925 b05 + -.48625778 b06 + .04122120 b07 + .03834568 b08 &
+ -.11416722 b09 + -.01779185 b10 + -.02334215 b11 + .19003009 a01 &
+ -.03353251 a02 + .04445021 a03 + -.22108040 a04 + .24306093 a05 &
+ -.01840429 a06 + .03016694 a07 + -.08137486 a08 + .10780627 a09 &
+ .00983544 a10 + .09080612 a11 + -.10410120 a12 + .08849923 a13 &
+ -.08163715 a14 + -.00448180 a15 + .00479276 a16 + -.01741007 t01 &
+ -.01527592 t02 + -.01491748 t03 + .00552530 t05 + .00679251 t08 &
+ .00965709 t09 + .01173937 t10 + .00773039 t11 + -.00563174 t12 &
+ -.00725649 t13 + -.00335937 t14
q005 = -.10997021 b01 + -.47330682 b02 + -.07048381 b03 + -.27664581 b04 &
+ .52618342 b05 + .49176753 b06 + .13554433 b07 + .02825867 b08 &
+ -.00174124 b09 + -.01632109 b10 + -.16376270 b11 + .02520455 a01 &
+ .00579795 a02 + .05901848 a03 + .12200625 a04 + -.05158626 a05 &
+ -.06255069 a06 + .02400442 a07 + .06579972 a08 + -.07134249 a09 &
+ .00814136 a10 + .11474421 a11 + -.18268419 a12 + .15428724 a13 &
+ -.09691134 a14 + .00809677 a15 + -.01787868 a16 + .01004980 t01 &
+ .00260137 t03 + .00383174 t04 + -.01175399 t05 + -.01204499 t06 &
+ -.00280651 t07 + .00536472 t08 + -.01094638 t09 + .01911303 t10 &
+ .01543182 t11 + .01635570 t12 + -.01351778 t13 + -.00363614 t14
q006 = -.05681351 b01 + -.23646515 b02 + -.07154246 b03 + .08336719 b04 &
+ -.45139779 b05 + .06810769 b06 + .77398568 b07 + .02065607 b08 &
+ -.05987140 b09 + -.00645249 b10 + .01266693 b11 + -.08361729 a01 &
+ -.01317246 a02 + .00582614 a03 + .00493928 a04 + .04840280 a05 &
+ -.04753391 a06 + .02225257 a07 + .00410554 a08 + .00701477 a09 &
+ .17575688 a10 + -.09653760 a11 + -.08072615 a12 + .05603803 a13 &
+ .11493927 a14 + -.19530350 a15 + -.00418691 a16 + .00110293 t01 &
+ -.00380374 t02 + .00983864 t03 + .00735398 t04 + -.00272936 t05 &
+ -.02717402 t06 + -.02006556 t07 + .01164925 t08 + -.01737971 t10 &
+ -.01440823 t11 + .00881363 t12 + .03467225 t13 + .01599178 t14
q007 = -.22620706 b01 + .10565612 b02 + -.01146724 b03 + -.40510869 b04 &
+ .09964628 b05 + -.33980248 b06 + .11849378 b07 + .61338312 b08 &
+ -.26868605 b09 + -.02696526 b10 + .02455669 b11 + -.12543285 a02 &
+ -.09934248 a03 + .05148357 a04 + .03067165 a05 + -.11198844 a06 &
+ .28535812 a07 + .03567612 a08 + .01071965 a09 + -.00733351 a10 &
+ .09019092 a11 + .01718209 a12 + -.14414116 a13 + .02010337 a14 &
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q020 = .06437422 b01 + .01371337 b02 + .03937275 b03 + .04084544 b04 &
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q021 = .01910839 b01 + .00938220 b02 + -.05248237 b03 + .00425694 b04 &
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End Of Internal Coordinates
FConstant
Square
21
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.299835636E-01 -.389772833E-02 -.108132508E-01 -.457534539E-02 -.629572430E-01 .788181076E-01
-.466921184E-02 .262121954E-02 .536441985E-01 .552303834E-02 -.420227065E-02 -.835912417E-02
.109288395E-02 .392323428E-02 .926839722E-02 .115269744E-01 .254987005E-02 -.553059359E-02
-.231041937E-02 .148017193E+00 .905480689E-03
.561288888E-02 -.691233565E-02 .114104286E-01 -.238410809E-02 -.545856308E-02 -.442161662E-02
-.437102328E-02 -.604754704E-02 .107752464E-02 -.833221471E-02 .217003076E-01 .951078339E-02
.561271000E-02 -.123280159E-01 -.227920332E-02 .523682320E-02 -.212180978E-04 -.357908064E-02
-.700685899E-03 .905480689E-03 .183552654E-01
End of input
Instead, the force constant matrix could have been read from file RLXOLD using the OLDForce keyword. In the present example we started from a favorable situation and geometry, therefore few iterations are enough to obtain a final CASSCF structure for the transition state. In any case it can be observed in the following extracts of the output file that close to the convergence criteria some problems with the convergence may appear. This is common for transition states, which have usually flat potential energy surfaces.
*************************************************************************************
************************** Energy Statistics for Geometry Optimization **************
*************************************************************************************
Energy Grad Grad Step Estimated Hessian
Iter Energy Change Norm Max Element Max Element Final Energy Index
1 -116.98834379 .00000000 .019829 .009618 Q007 .101378 Q002 -116.98789095 1
2 -116.98875987 -.00041608 .014947 .004894 Q002 .066764 Q007 -116.98841151 1
3 -116.98841391 .00034596 .001938 .000775 Q008 .005446 Q008 -116.98841751 1
4 -116.98841831 -.00000440 .000514 .000172 Q008 -.001494 Q021 -116.98841867 1
Cartesian Displacement Gradient (Int. Coord.)
Value Threshold Converged? Value Threshold Converged?
+----------------------------------+----------------------------------+
RMS + .1188E-02 .1200E-02 Yes + .1149E-03 .3000E-03 Yes +
+----------------------------------+----------------------------------+
Max + .1560E-02 .1800E-02 Yes + .1724E-03 .4500E-03 Yes +
+----------------------------------+----------------------------------+
Geometry is converged
*************************************************************************************
*************************************************************************************
Table 4.17 compiles the obtained CASSCF geometries for reactant, transition state, and product in the present example. They can be compared to the MP2 geometries [50]. The overall agreement is good. The wave function at each of the geometries was proved to be almost a single configuration. The second configuration in all the cases contributed by less than 5% to the weight of the wave function. It is a double excited replacement. Therefore, although MP2 is not generally expected to describe properly a bond formation in this case its behavior seems to be validated. The larger discrepancies appear in the carbon-carbon distances in the dimethylcarbene and in the transition state. On one hand the basis set used in the present example were small; on the other hand there are indications that the MP2 method overestimates the hyper conjugation effects present in the dimethylcarbene [50]. Figure 4.8 displays the dimethylcarbene with indication of the employed labeling.
CC |
CC |
CCC |
CCH |
CCCH |
CH |
CH |
CCH |
CCCH |
|
| Dimethylcarbene | |||||||||
CAS |
1.497 | 1.497 | 110.9 | 102.9 | 88.9 | 1.099 | 102.9 | 88.9 | |
MP2 |
1.480 | 1.480 | 110.3 | 98.0 | 85.5 | 1.106 | 98.0 | 85.5 | |
| Transition structure | |||||||||
| CAS |
1.512 | 1.394 | 114.6 | 106.1 | 68.6 | 1.287 | 1.315 | 58.6 | 76.6 |
| MP2 |
1.509 | 1.402 | 112.3 | 105.1 | 69.2 | 1.251 | 1.326 | 59.6 | 77.7 |
| Propene | |||||||||
| CAS |
1.505 | 1.344 | 124.9 | 110.7 | 59.4 | ||||
| MP2 |
1.501 | 1.338 | 124.4 | 111.1 | 59.4 | ||||
 C, carbenoid center; C, carbon which looses the hydrogen H. See Figure 4.8. |
|||||||||
Present results. CASSCF, ANO-S C , H  . Two electrons in two orbitals. |
|||||||||
| MP2 6-31G(2p,d), Ref. [50]. |
|||||||||
The main structural effects occurring during the reaction can be observed
displayed in Table 4.17. As the rearrangement starts out one
hydrogen atom (H) moves in a plane almost perpendicular to the plane
formed by the three carbon atoms while the remaining two hydrogen atoms
on the same methyl group swing very rapidly into a nearly planar
position (see Figure 4.7 on page ![[*]](crossref.gif)
).
As the 
bond is formed we observe
a contraction of the C-C distance. In contrast, the spectator
methyl group behaves as a rigid body. Their parameters were
not compiled here but it rotates and bends slightly [50]. Focusing
on the second half reaction, the moving hydrogen atom rotates into the
plane of the carbon atoms to form the new C-H bond. This movement
is followed by a further shortening of the preformed C-C bond,
which adquires the bond distance of a typical double carbon bond, and
smaller adjustments in the positions of the other atoms. The structures
of the reactant, transition state, and product are shown in
Figure 4.7.
As was already mentioned we will apply now higher-correlated methods for the reactant, product, and transition state system at the CASSCF optimized geometries to account for more accurate relative energies. In any case a small basis set has been used and therefore the goal is not to be extremely accurate. For more complete results see Ref. [50]. We are going to perform calculations with the MP2, MRCI, ACPF, CASPT2, CCSD, and CCSD(T) methods.
Starting with dimethylcarbene, we will use the following input file together with the AUTOMOLCAS program.
&SEWARD &END Title Dimethylcarbene singlet C2-sym CASSCF(ANO-VDZP) opt geometry Symmetry XY Basis set C.ANO-S...3s2p1d. C1 .0000000000 .0000000000 1.2019871414 C2 .0369055124 2.3301037548 -.4006974719 End of basis Basis set H.ANO-S...2s1p. H1 -.8322309260 2.1305589948 -2.2666729831 H2 -.7079699536 3.9796589218 .5772009623 H3 2.0671154914 2.6585385786 -.6954193494 End of basis PkThrs 1.0E-10 End of input &SCF &END Title Dmc Occupied 7 5 End of input &RASSCF &END Title Dmc Symmetry 1 Spin 1 Nactel 2 0 0 Inactive 6 5 Ras2 1 1 Thrs 1.0E-05,1.0E-03,1.0E-03 Iteration 50,25 LumOrb End of input &CASPT2 &END Title Dmc LRoot 1 Frozen 2 1 End of input &MOTRA &END Title Dmc Frozen 2 1 JobIph End of input &GUGA &END Title Dmc Electrons 18 Spin 1 Symmetry 2 Inactive 4 4 Active 1 1 Ciall 1 Print 5 End of input &MRCI &END Title Dimethylcarbene SDCI End of input &MRCI &END Title Dimethylcarbene ACPF End of input * Now we generate the single ref. function * for coupled-cluster calculations &RASSCF &END Title Dmc Symmetry 1 Spin 1 Nactel 0 0 0 Inactive 7 5 Ras2 0 0 Thrs 1.0E-05,1.0E-03,1.0E-03 Iteration 50,25 LumOrb Canonical End of input &MOTRA &END Title Dmc Frozen 2 1 JobIph End of input &CCSORT &END Title Dmc CCT End of input &CCSD &END Title Dmc Iterations 40 Triples 2 End of input &CCT3 &END Title Dmc Triples 2 End of input
To run AUTOMOLCAS we will use the script:
#!/bin/ksh export MOLCAS=/a/molcas.4.0 export Project=dmc export HomeDir=$PWD export WorkDir=/temp/$LOGNAME/$Project mkdir $WorkDir cd $WorkDir molcas run automolcas $HomeDir/$Project.input cd - rm -r $WorkDir exit
Observe in the previous input that we have generated a multiconfigurational wave function for CASPT2, MRCI, and ACPF wave functions but a single configuration reference wave function (using RASSCF program with the option CANOnical) for the CCSD and CCSD(T) wave functions. Notice also that to compute a multiconfigurational ACPF wave function we have to use the MRCI program, not the CPF module which does not accept more than one single reference. In all the highly correlated methods we have frozen the three carbon core orbitals because of the reasons already explained in section 4.1. For MRCI, ACPF, CCSD, and CCSD(T) the freezing is performed in the MOTRA step.
One question that can be addressed is which is the proper reference space
for the multiconfigurational calculations. As was explained when we selected
the active space for the geometry optimizations, we performed several tests
at different stages in the reaction path and observed that the smallest
meaningful active space, two electrons in two orbitals, was sufficient
in all the cases. We can come back to this problem here to select the
reference for CASPT2, MRCI, and ACPF methods. The simple analysis of the
SCF orbital energies shows that in dimethylcarbene, for instance, the
orbital energies of the C-H bonds are close to those of the C-C 
bonds and additionally those orbitals are strongly mixed along
the reaction path. A balanced active space including all orbitals necessary
to describe the shifting H-atom properly would require a full valence
space of 18 electrons in 18 orbitals. This is not a feasible space, therefore
we proceed with the minimal active space and analyze later the quality
of the results. The CASSCF wave function will then include for dimethylcarbene
and the transition state structure the ()
()
and
()() configurations correlating the non-bonded electrons
localized at the carbenoid center where as for propene the active space
include the equivalent valence space.
The GUGA input must be built carefully. There are several ways to specify the reference configurations for the following methods. First, the keyword ELECtrons refers to the total number of electrons that are going to be correlated, that is, all except those frozen in the previous MOTRA step. Keyword SYMMetry refers not to the number of the symmetry of the resulting wave function but to the total number of symmetries of the employed point group. Keywords INACtive and ACTIve are optional and describe the number of inactive (occupation two in all the reference configurations) and active (varying occupation number in the reference configurations) orbitals of the space. Here ACTIve indicates one orbital of each of the symmetries. The following keyword CIALl indicates that the reference space will be the full CI within the subspace of active orbitals. It must be always followed by symmetry index (number of symmeteries) for the resulting wave function, one here.
For the transition state structure we do not impose any symmetry
restriction, therefore the calculations are performed in the C group
with the input file:
&SEWARD &END Title Dimethylcarbene to propene Transition State C1 symmetry CASSCF (ANO-VDZP) opt geometry Basis set C.ANO-S...3s2p1d. End of basis Basis set H.ANO-S...2s1p. End of basis PkThrs 1.0E-10 End of input &SCF &END Title Ts Occupied 12 End of input &MBPT2 &END Title Ts Frozen 3 End of input &RASSCF &END Title Ts Symmetry 1 Spin 1 Nactel 2 0 0 Inactive 11 Ras2 2 Iteration 50,25 LumOrb End of input &CASPT2 &END Title Ts LRoot 1 Frozen 3 End of input &MOTRA &END Title Ts Frozen 3 JobIph End of input &GUGA &END Title Ts Electrons 18 Spin 1 Symmetry 1 Inactive 8 Active 2 Ciall 1 Print 5 End of input &MRCI &END Title Ts SDCI End of input &MRCI &END Title Ts ACPF End of input &RASSCF &END Title Ts Symmetry 1 Spin 1 Nactel 0 0 0 Inactive 12 Ras2 0 Iteration 50,25 LumOrb Canonical End of input &MOTRA &END Title Ts Frozen 3 JobIph End of input &CCSORT &END Title Ts CCT End of input &CCSD &END Title Ts Iterations 40 Triples 2 End of input &CCT3 &END Title Ts Triples 2 End of input
Finally we compute the wave functions for the product, propene, in the
C symmetry group with the input:
&SEWARD &END Title Propene singlet Cs-sym CASSCF(ANO-VDZP) opt geometry Symmetry Z Basis set C.ANO-S...3s2p1d. C1 -2.4150580342 .2276105054 .0000000000 C2 .0418519070 .8733601069 .0000000000 C3 2.2070668305 -.9719171861 .0000000000 End of basis Basis set H.ANO-S...2s1p. H1 -3.0022907382 -1.7332097498 .0000000000 H2 -3.8884900111 1.6454331428 .0000000000 H3 .5407865292 2.8637419734 .0000000000 H4 1.5296107561 -2.9154199848 .0000000000 H5 3.3992878183 -.6985812202 1.6621549148 End of basis PkThrs 1.0E-10 End of input &SCF &END Title Propene Occupied 10 2 End of input &MBPT2 &END Title Propene Frozen 3 0 End of input &RASSCF &END Title Propene Symmetry 1 Spin 1 Nactel 2 0 0 Inactive 10 1 Ras2 0 2 Thrs 1.0E-05,1.0E-03,1.0E-03 Iteration 50,25 LumOrb End of input &CASPT2 &END Title Propene LRoot 1 Frozen 3 0 End of input &MOTRA &END Title Propene Frozen 3 0 JobIph End of input &GUGA &END Title Propene Electrons 18 Spin 1 Symmetry 2 Inactive 7 1 Active 0 2 Ciall 1 Print 5 End of input &MRCI &END Title Propene SDCI End of input &MRCI &END Title Propene ACPF End of input &RASSCF &END Title Propene Symmetry 1 Spin 1 Nactel 0 0 0 Inactive 10 2 Ras2 0 0 Thrs 1.0E-05,1.0E-03,1.0E-03 Iteration 50,25 LumOrb Canonical End of input &MOTRA &END Title Propene Frozen 3 0 JobIph End of input &CCSORT &END Title Propene CCT End of input &CCSD &END Title Propene Iterations 40 Triples 2 End of input &CCT3 &END Title Propene Triples 2 End of input
Table 4.18 compiles the total and relative energies obtained for the studied reaction at the different levels of theory employed.
| Single configurational methods | ||||
| method | RHF | MP2 | CCSD | CCSD(T) |
| Dimethylcarbene | ||||
| -117.001170 | -117.392130 | -117.442422 | -117.455788 | |
| Transition state structure | ||||
| -116.972670 | -117.381342 | -117.424088 | -117.439239 | |
| BH |
(17.88) | (6.77) | (11.50) | (10.38) |
| Propene | ||||
| -117.094700 | -117.504053 | -117.545133 | -117.559729 | |
| EX |
(-58.69) | (-70.23) | (-64.45) | (-65.22) |
| Multiconfigurational methods | ||||
| method | CASSCF | CASPT2 | SD-MRCI+Q | ACPF |
| Dimethylcarbene | ||||
| -117.020462 | -117.398025 | -117.447395 | -117.448813 | |
| Transition state structure | ||||
| -116.988419 | -117.383017 | -117.430951 | -117.432554 | |
| BH |
(20.11) | (9.42) | (10.32) | (10.20) |
| Propene | ||||
| -117.122264 | -117.506315 | -117.554048 | -117.554874 | |
| EX |
(-63.88) | (-67.95) | (-66.93) | (-66.55) |
| Barrier height. Needs to be corrected with the zero point vibrational correction. |
||||
| Exothermicity. Needs to be corrected with the zero point vibrational correction. |
||||
We can discuss now the quality of the results obtained and their
reliability (for a more careful discusion of the accuracy of
quantum chemical calculations see Ref. [13]).
In first place we have to consider that a valence
double-zeta plus polarization basis set is somewhat small to obtain
accurate results. At least a triple-zeta quality would be required.
The present results have, however, the goal to serve as an example.
We already pointed out that the CASSCF geometries were very similar
to the MP2 reported geometries [50]. This fact validates
both methods. MP2 provides remarkably accurate geometries using
basis sets of triple-zeta quality, as in Ref. [50], in
situations were the systems can be described as singly configurational,
as the CASSCF calculations show. The Hartree-Fock configuration has
a contribution of more than 95% in all three structures, while the
largest weight for another configuration appears in propene for
()(
) (4.2%).
The MRCI calculations provide also one test of the validity of the reference wave function. For instance, the MRCI output for propene is:
FINAL RESULTS FOR STATE NR 1
CORRESPONDING ROOT OF REFERENCE CI IS NR: 1
REFERENCE CI ENERGY: -117.12226386
EXTRA-REFERENCE WEIGHT: .11847074
CI CORRELATION ENERGY: -.38063043
CI ENERGY: -117.50289429
DAVIDSON CORRECTION: -.05115380
CORRECTED ENERGY: -117.55404809
ACPF CORRECTION: -.04480105
CORRECTED ENERGY: -117.54769535
CI-COEFFICIENTS LARGER THAN .050
NOTE: THE FOLLOWING ORBITALS WERE FROZEN
ALREADY AT THE INTEGRAL TRANSFORMATION STEP
AND DO NOT EXPLICITLY APPEAR:
SYMMETRY: 1 2
PRE-FROZEN: 3 0
ORDER OF SPIN-COUPLING: (PRE-FROZEN, NOT SHOWN)
(FROZEN, NOT SHOWN)
VIRTUAL
ADDED VALENCE
INACTIVE
ACTIVE
ORBITALS ARE NUMBERED WITHIN EACH SEPARATE SYMMETRY.
CONFIGURATION 32 COEFFICIENT -.165909 REFERENCE
SYMMETRY 1 1 1 1 1 1 1 2 2 2
ORBITALS 4 5 6 7 8 9 10 1 2 3
OCCUPATION 2 2 2 2 2 2 2 2 0 2
SPIN-COUPLING 3 3 3 3 3 3 3 3 0 3
CONFIGURATION 33 COEFFICIENT -.000370 REFERENCE
SYMMETRY 1 1 1 1 1 1 1 2 2 2
ORBITALS 4 5 6 7 8 9 10 1 2 3
OCCUPATION 2 2 2 2 2 2 2 2 1 1
SPIN-COUPLING 3 3 3 3 3 3 3 3 1 2
CONFIGURATION 34 COEFFICIENT .924123 REFERENCE
SYMMETRY 1 1 1 1 1 1 1 2 2 2
ORBITALS 4 5 6 7 8 9 10 1 2 3
OCCUPATION 2 2 2 2 2 2 2 2 2 0
SPIN-COUPLING 3 3 3 3 3 3 3 3 3 0
**************************************************************
The Hartree-Fock configuration contributes to the MRCI configuration with a weight of 85.4%, while the next configuration contributes by 2.8%. Similar conclusions can be obtained analyzing the ACPF results and for the other structures. We will keep the MRCI results including the Davidson correction (MRCI+Q) which corrects for the size-inconsistency of the truncated CI expansion [13].
For CASPT2 the evaluation criteria were already commented in section 4.3. The portion of the CASPT2 output for propene is:
Reference energy: -117.1222638304
E2 (Non-variational): -.3851719971
E2 (Variational): -.3840516039
Total energy: -117.5063154343
Residual norm: .0000000000
Reference weight: .87905
Contributions to the CASPT2 correlation energy
Active & Virtual Only: -.0057016698
One Inactive Excited: -.0828133881
Two Inactive Excited: -.2966569393
----------------------------------------------------------------------------
Report on small energy denominators, large components, and large energy contributions.
The ACTIVE-MIX index denotes linear combinations which gives ON expansion functions
and makes H0 diagonal within type.
DENOMINATOR: The (H0_ii - E0) value from the above-mentioned diagonal approximation.
RHS value: Right-Hand Side of CASPT2 Eqs.
COEFFICIENT: Multiplies each of the above ON terms in the first-order wave function.
Thresholds used:
Denominators: .3000
Components: .0250
Energy contributions: .0050
CASE SYMM ACTIVE NON-ACT IND DENOMINATOR RHS VALUE COEFFICIENT CONTRIBUTION
AIVX 1 Mu1.0003 In1.004 Se1.022 2.28926570 .05988708 -.02615995 -.00156664
The weight of the CASSCF reference to the first-order wave function is here 87.9%, very close to the weights obtained for the dimethylcarbene and the transition state structure, and there is only a small contribution to the wave function and energy which is larger than the selected thresholds. This should not be considered as a intruder state, but as a contribution from the fourth inactive orbital which could be, eventually, included in the active space. The contribution to the second-order energy in this case is smaller than 1 Kcal/mol. It can be observed that the same contribution shows up for the transition state structure but not for the dimethylcarbene. In principle this could be an indication that a larger active space, that is, four electrons in four orbitals, would give a slightly more accurate CASPT2 energy. The present results will probably overestimate the second-order energies for the transition state structure and the propene, leading to a slightly smaller activation barrier and a slightly larger exothermicity, as can be observed in Table 4.18. The orbitals pointed out as responsible for the large contributions in propene are the fourth inactive and 22nd secondary orbitals of the first symmetry. They are too deep and too high, respectively, to expect that an increase in the active space could in fact represent a great improvement in the CASPT2 result. In any case we tested for four orbitals-four electrons CASSCF/CASPT2 calculations and the results were very similar to those presented here.
Finally we can analyze the so-called 
-diagnostic [55]
for the coupled-cluster wave functions. is defined for closed-shell
coupled-cluster methods as the Euclidian norm of the vector of T
amplitudes normalized by the number of electrons correlated:
.
In the output of the CCSD program we have:
Convergence after 17 Iterations
Total energy (diff) : -117.54513288 -.00000061
Correlation energy : -.45043295
E1aa contribution : .00000000
E1bb contribution : .00000000
E2aaaa contribution : -.04300448
E2bbbb contribution : -.04300448
E2abab contribution : -.36442400
Five largest amplitudes of :T1aa
SYMA SYMB SYMI SYMJ A B I J VALUE
2 0 2 0 4 0 2 0 -.0149364994
2 0 2 0 2 0 2 0 .0132231037
2 0 2 0 8 0 2 0 -.0104167047
2 0 2 0 7 0 2 0 -.0103366543
2 0 2 0 1 0 2 0 .0077537734
Euclidian norm is : .0403635306
Five largest amplitudes of :T1bb
SYMA SYMB SYMI SYMJ A B I J VALUE
2 0 2 0 4 0 2 0 -.0149364994
2 0 2 0 2 0 2 0 .0132231037
2 0 2 0 8 0 2 0 -.0104167047
2 0 2 0 7 0 2 0 -.0103366543
2 0 2 0 1 0 2 0 .0077537734
Euclidian norm is : .0403635306
In this case T1aa and T1bb are identical because we are computing a
closed-shell singlet state. The five largest T amplitudes are
printed, as well as the Euclidian norm. Here the number of correlated
electrons is 18, therefore the value for the diagnostic is 0.01.
This value can be considered acceptable as evaluation of the
quality of the calculation. The use of as a diagnostic is
based on an observed empirical correlation: larger values give poor
CCSD results for molecular structures, binding energies, and
vibrational frequencies [56]. It was considered that values
larger than 0.02 indicated that results from single-reference electron
correlation methods limited to single and double excitations should be
viewed with caution.
There are several considerations concerning the diagnostic
[55]. First, it is only valid within the frozen core
approximation and it was defined for coupled-cluster procedures
using SCF molecular orbitals in the reference function. Second, it is
a measure of the importance of non-dynamical electron correlation effects
and not of the degree of the multireference effects. Sometimes the two
effects are related, but not always (see discusion in Ref. [56]).
Finally, the performance of the CCSD(T) method is reasonably good even
in situations where has a value as large as 0.08.
In conclusion, the use of together with other wave function
analysis, such as explicitely examining the largest T and T
amplitudes, is the best approach to evaluate the quality of the
calculations but this must be done with extreme caution.
As the present systems are reasonably well described by a single
determinant reference function there is no doubt that the CCSD(T)
method provides the most accurate results. Here CASPT2, MRCI+Q,
ACPF, and CCSD(T) predict the barrier height from the reactant
to the transition state with an accuracy better than 1 Kcal/mol.
The correspondence is somewhat worse, about 3 Kcal/mol, for the
exothermicity. As the difference is largest for the CCSD(T) method
we may conclude than triple and higher order excitations are of
importance to achieve a balanced correlation treatment, in particular
with respect to the partially occupied orbital at the
carbenoid center. It is also noticeable that the relative MP2
energies appear to be shifted about 3-4 Kcal/mol towards lower
values. This effect may be due to the overestimation of the
hyperconjugation effect which appears to be strongest in dimethylcarbene
[51,50].
Additional factors affecting the accuracy of the results obtained are the zero point vibrational energy correction and, of course, the saturation of the one particle basis sets. The zero point vibrational correction could be computed by performing a numerical harmonic vibrational analysis at the CASSCF level using MOLCAS. At the MP2 level [50] the obtained values were -1.1 Kcal/mol and 2.4 Kcal/mol for the activation barrier height and exothermicity, respectively. Therefore, if we take as our best values the CCSD(T) results of 10.4 and -65.2 Kcal/mol, respectively, our prediction would be an activation barrier height of 9.3 Kcal/mol and an exothermicity of -62.8 Kcal/mol. Calculations with larger basis sets and MP2 geometries gave 7.4 and -66.2 Kcal/mol, respectively [50]. The experimental estimation gives a lower limit to the activation barrier of 3.3 Kcal/mol [50].
MOLCAS provides also a number of one-electron properties which can be useful to analyze the chemical behavior of the systems. For instance, the Mulliken population analysis is available for the RHF, CASSCF, CASPT2, MRCI, and ACPF wave functions. Mulliken charges are known to be strongly biased by the choice of the basis sets, nevertheless one can restrict the analysis to the relative charge differences during the course of the reaction to obtain a qualitative picture. We can use, for instance, the charge distribution obtained for the MRCI wave funcion, which is listed in Table 4.19. Take into account that the absolute values of the charges can vary with the change of basis set.
C |
C |
H |
 |
H+H |
Me |
||
| Dimethylcarbene | |||||||
| -0.12 | -0.13 | 0.05 | -0.20 | 0.14 | 0.07 | ||
| Transition state structure | |||||||
| -0.02 | -0.23 | 0.05 | -0.20 | 0.17 | 0.02 | ||
| Propene | |||||||
| -0.18 | -0.02 | 0.05 | -0.15 | 0.18 | -0.02 | ||
| Carbon from which the hydrogen is withdrawn. |
|||||||
| Central carbenoid carbon. |
|||||||
| Migrating hydrogen. |
|||||||
 Sum of charges for centers C, C, and H. |
|||||||
 Sum of charges for the remaining hydrogens attached to C. |
|||||||
| Sum of charges for the spectator methyl group. |
|||||||
In dimethylcarbene both the medium and terminal carbons appear equally charged.
During the migration of hydrogen H charge flows from the hydrogen donating
carbon, C, to the carbenoid center. For the second half of the reaction
the charge flows back to the terminal carbon from the centered carbon, probably
due to the effect of the delocalization.