5.1.4. High quality wave functions at optimized structures¶
Here we will give an example of how geometrical structures obtained at one level of theory can be used in an analysis at high quality wave functions. Table 5.1.4.1 compiles the obtained CASSCF geometries for the dimethylcarbene to propene reaction (see Figure 5.1.4.1). They can be compared to the MP2 geometries [282]. 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 carboncarbon 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 [282]. Figure 5.1.4.2 displays the dimethylcarbene with indication of the employed labeling.

\(\ce{C{1}C{3}}\) 
\(\ce{C{1}C{2}}\) 
\(\ce{C{2}C{1}C{3}}\) 
\(\ce{C{1}C{3}H{6}}\) 
\(\ce{C{2}C{1}C{3}H{6}}\) 
\(\ce{C{2}H{5}}\) 
\(\ce{C{1}H{5}}\) 
\(\ce{C{1}C{2}H{5}}\) 
\(\ce{C{3}C{1}C{2}H{5}}\) 

Dimethylcarbene 

CAS2 
1.497 
1.497 
110.9 
102.9 
88.9 
1.099 
102.9 
88.9 

MP23 
1.480 
1.480 
110.3 
98.0 
85.5 
1.106 
98.0 
85.5 

Transition structure 

CAS2 
1.512 
1.394 
114.6 
106.1 
68.6 
1.287 
1.315 
58.6 
76.6 
MP23 
1.509 
1.402 
112.3 
105.1 
69.2 
1.251 
1.326 
59.6 
77.7 
Propene 

CAS2 
1.505 
1.344 
124.9 
110.7 
59.4 

MP23 
1.501 
1.338 
124.4 
111.1 
59.4 
 1
\(\ce{C{1}}\), carbenoid center; \(\ce{C{2}}\), carbon which loses the hydrogen \(\ce{H{5}}\). See Figure 5.1.4.2.
 2(1,2,3)
Present results. CASSCF, ANOS C[3s2p1d], H[2d1p]. Two electrons in two orbitals.
 3(1,2,3)
MP2 631G(2p,d), Ref. [282].
The main structural effects occurring during the reaction can be observed displayed in Table 5.1.4.1. As the rearrangement starts out one hydrogen atom (\(\ce{H{5}}\)) 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 5.1.4.1). As the \(\pi\) bond is formed we observe a contraction of the \(\ce{C{1}C{2}}\) distance. In contrast, the spectator methyl group behaves as a rigid body. Their parameters were not compiled here but it rotates and bends slightly [282]. Focusing on the second half reaction, the moving hydrogen atom rotates into the plane of the carbon atoms to form the new \(\ce{C{1}H{5}}\) bond. This movement is followed by a further shortening of the preformed \(\ce{C{1}C{2}}\) bond, which acquires 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 5.1.4.1.
As was already mentioned we will apply now highercorrelated 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. [282]. 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:
&SEWARD &END
Title
Dimethylcarbene singlet C2sym
CASSCF(ANOVDZP) opt geometry
Symmetry
XY
Basis set
C.ANOS...3s2p1d.
C1 .0000000000 .0000000000 1.2019871414
C2 .0369055124 2.3301037548 .4006974719
End of basis
Basis set
H.ANOS...2s1p.
H1 .8322309260 2.1305589948 2.2666729831
H2 .7079699536 3.9796589218 .5772009623
H3 2.0671154914 2.6585385786 .6954193494
End of basis
PkThrs
1.0E10
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
1.0E05,1.0E03,1.0E03
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
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 coupledcluster calculations
&RASSCF &END
Title
Dmc
Symmetry
1
Spin
1
Nactel
0 0 0
Inactive
7 5
Ras2
0 0
Thrs
1.0E05,1.0E03,1.0E03
Iteration
50,25
LumOrb
OutOrbitals
Canonical
End of input
&MOTRA &END
Title
Dmc
Frozen
2 1
JobIph
End of input
&CCSDT &END
Title
Dmc
CCT
Iterations
40
Triples
2
End of input
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 options OUTOrbitals and 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 5.1.1. 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 \(\ce{CH}\) bonds are close to those of the \(\ce{CC}\) \(\sigma\) bonds and additionally those orbitals are strongly mixed along the reaction path. A balanced active space including all orbitals necessary to describe the shifting Hatom 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 \((\sigma)^2(\pi)^0\) and \((\sigma)^0(\pi)^2\) configurations correlating the nonbonded electrons localized at the carbenoid center where as for propene the active space include the equivalent valence \(\pi\) 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. 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 the irrep) 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 \(C_1\) group with the input file:
&SEWARD &END
Title
Dimethylcarbene to propene
Transition State C1 symmetry
CASSCF (ANOVDZP) opt geometry
Basis set
C.ANOS...3s2p1d.
End of basis
Basis set
H.ANOS...2s1p.
End of basis
PkThrs
1.0E10
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
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
OutOrbitals
Canonical
End of input
&MOTRA &END
Title
Ts
Frozen
3
JobIph
End of input
&CCSDT &END
Title
Ts
CCT
Iterations
40
Triples
2
End of input
Finally we compute the wave functions for the product, propene, in the \(C_s\) symmetry group with the input:
&SEWARD &END
Title
Propene singlet Cssym
CASSCF(ANOVDZP) opt geometry
Symmetry
Z
Basis set
C.ANOS...3s2p1d.
C1 2.4150580342 .2276105054 .0000000000
C2 .0418519070 .8733601069 .0000000000
C3 2.2070668305 .9719171861 .0000000000
End of basis
Basis set
H.ANOS...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
1.0E10
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
1.0E05,1.0E03,1.0E03
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
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
1.0E05,1.0E03,1.0E03
Iteration
50,25
LumOrb
OutOrbitals
Canonical
End of input
&MOTRA &END
Title
Propene
Frozen
3 0
JobIph
End of input
&CCSDT &END
Title
Propene
CCT
Iterations
40
Triples
2
End of input
Table 5.1.4.2 compiles the total and relative energies obtained for the studied reaction at the different levels of theory employed.
Single configurational methods 



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 

BH4 
(17.88) 
(6.77) 
(11.50) 
(10.38) 
Propene 

−117.094700 
−117.504053 
−117.545133 
−117.559729 

EX5 
(−58.69) 
(−70.23) 
(−64.45) 
(−65.22) 
Multiconfigurational methods 



CASSCF 
CASPT2 
SDMRCI+Q 
ACPF 
Dimethylcarbene 

−117.020462 
−117.398025 
−117.447395 
−117.448813 

Transition state structure 

−116.988419 
−117.383017 
−117.430951 
−117.432554 

BH4 
(20.11) 
(9.42) 
(10.32) 
(10.20) 
Propene 

−117.122264 
−117.506315 
−117.554048 
−117.554874 

EX5 
(−63.88) 
(−67.95) 
(−66.93) 
(−66.55) 
We can discuss now the quality of the results obtained and their reliability (for a more careful discussion of the accuracy of quantum chemical calculations see Ref. [271]). In first place we have to consider that a valence doublezeta plus polarization basis set is somewhat small to obtain accurate results. At least a triplezeta 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 [282]. This fact validates both methods. MP2 provides remarkably accurate geometries using basis sets of triplezeta quality, as in Ref. [282], 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 \((\pi)^0(\pi^*)^2\) (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
EXTRAREFERENCE WEIGHT: .11847074
CI CORRELATION ENERGY: .38063043
CI ENERGY: 117.50289429
DAVIDSON CORRECTION: .05115380
CORRECTED ENERGY: 117.55404809
ACPF CORRECTION: .04480105
CORRECTED ENERGY: 117.54769535
CICOEFFICIENTS LARGER THAN .050
NOTE: THE FOLLOWING ORBITALS WERE FROZEN
ALREADY AT THE INTEGRAL TRANSFORMATION STEP
AND DO NOT EXPLICITLY APPEAR:
SYMMETRY: 1 2
PREFROZEN: 3 0
ORDER OF SPINCOUPLING: (PREFROZEN, 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
SPINCOUPLING 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
SPINCOUPLING 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
SPINCOUPLING 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 sizeinconsistency of the truncated CI expansion [271].
For CASPT2 the evaluation criteria are commented in Section 5.1.5. The portion of the CASPT2 output for propene is:
Reference energy: 117.1222638304
E2 (Nonvariational): .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 ACTIVEMIX index denotes linear combinations which gives ON expansion functions
and makes H0 diagonal within type.
DENOMINATOR: The (H0_ii  E0) value from the abovementioned diagonal approximation.
RHS value: RightHand Side of CASPT2 Eqs.
COEFFICIENT: Multiplies each of the above ON terms in the firstorder wave function.
Thresholds used:
Denominators: .3000
Components: .0250
Energy contributions: .0050
CASE SYMM ACTIVE NONACT 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 firstorder 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 secondorder 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 secondorder 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 5.1.4.2. 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 orbitalsfour electrons CASSCF/CASPT2 calculations and the results were very similar to those presented here.
Finally we can analyze the socalled \(\tau_1\)diagnostic [283] for the coupledcluster wave functions. \(\tau_1\) is defined for closedshell coupledcluster methods as the Euclidean norm of the vector of \(T_1\) amplitudes normalized by the number of electrons correlated: \(\tau_1 = \lVert T_1\Vert/N_{\text{el}}^{1/2}\). 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
Euclidean 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
Euclidean norm is : .0403635306
In this case T1aa and T1bb are identical because we are computing a closedshell singlet state. The five largest \(T_1\) amplitudes are printed, as well as the Euclidean norm. Here the number of correlated electrons is 18, therefore the value for the \(\tau_1\) diagnostic is 0.01. This value can be considered acceptable as evaluation of the quality of the calculation. The use of \(\tau_1\) as a diagnostic is based on an observed empirical correlation: larger values give poor CCSD results for molecular structures, binding energies, and vibrational frequencies [284]. It was considered that values larger than 0.02 indicated that results from singlereference electron correlation methods limited to single and double excitations should be viewed with caution.
There are several considerations concerning the \(\tau_1\) diagnostic [283]. First, it is only valid within the frozen core approximation and it was defined for coupledcluster procedures using SCF molecular orbitals in the reference function. Second, it is a measure of the importance of nondynamical electron correlation effects and not of the degree of the multireference effects. Sometimes the two effects are related, but not always (see discussion in Ref. [284]). Finally, the performance of the CCSD(T) method is reasonably good even in situations where \(\tau_1\) has a value as large as 0.08. In conclusion, the use of \(\tau_1\) together with other wave function analysis, such as explicitly examining the largest \(T_1\) and \(T_2\) 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 \(\pi^*\) 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 [285][282].
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 [282] 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 [282]. The experimental estimation gives a lower limit to the activation barrier of 3.3 kcal/mol [282].
Molcas provides also a number of oneelectron 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 function, which is listed in Table 5.1.4.5. Take into account that the absolute values of the charges can vary with the change of basis set.
\(\ce{C{2}}\)6 
\(\ce{C{1}}\)7 
\(\ce{H{5}}\)8 
\(\Sigma\)9 
\(\ce{H{1}}+\ce{H{3}}\)10 
\(\ce{Me}\)11 

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 
 6
Carbon from which the hydrogen is withdrawn.
 7
Central carbenoid carbon.
 8
Migrating hydrogen.
 9
Sum of charges for centers \(\ce{C{2}}\), \(\ce{C{1}}\), and \(\ce{H{5}}\).
 10
Sum of charges for the remaining hydrogens attached to \(\ce{C{2}}\).
 11
Sum of charges for the spectator methyl group.
In dimethylcarbene both the medium and terminal carbons appear equally charged. During the migration of hydrogen \(\ce{H{5}}\) charge flows from the hydrogen donating carbon, \(\ce{C{2}}\), 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 \(\pi\) delocalization.