The computed states will use different partitionings of the active space. The basic valence space (1302) must be included in all the cases. The valence states only involve excitations into the and orbitals. Therefore they belong to the A_{1} and B_{2} symmetries. In addition we can have single excitations (Rydberg states) from the occupied orbitals to the Rydberg orbitals of b_{1} and a_{2} symmetries. The number of Rydberg orbitals belonging to those symmetries is (0201). Thus, the final space to compute simultaneously valence and Rydberg states is (1302) + (0201): (1503). The same space can be used to compute states because the n orbital and the orbitals are included into the active space. The symmetries of these states, however, will be A_{2} and B_{1}. In the table we also have another division for the A_{2} and B_{1}, R(), and A_{1}, R(), (only the 3s) Rydberg states, using an active space (5322). We have, therefore, divided the excited states to be computed by symmetries and active space. Stateaverage CASSCF calculations for each one of the cases have to be performed. The only question which remains is how many roots we have to include in each of the cases. This is also determined by the symmetry and active space available. For instance, for the A_{1} states, we want to compute the ground state plus three Rydberg states (see Table in both HOMO and HOMO1 n=3 series) plus a certain number of valence states. If we do not have any previous experience we may think of three or four possible valence states but we know that the usual number of lowlying valence states is close to the number of valence singly excited states, in this case two of A_{1} symmetry. This does not mean that the states are going to be described by one single configuration; it is simply an estimation of the number of relevant states based on experience. In summary, we expect to compute six A_{1} states and therefore we include six roots in the CASSCF stateaverage input. It is not uncommon that one or more valence states do not appear in the initial CASSCF calculation including the desired roots and other higher Rydberg states. This is due to the fact that valence states usually require larger dynamical correlation corrections than the Rydberg states. Therefore in a CASSCF calculation the Rydberg states are, in general, lower in energy than the valence states. The dynamical correlation included by the CASPT2 method will place the states correctly. However this is only possible if the states are present in the CASSCF calculation. It is then necessary to be sure that the states are located at the CASSCF level. Maybe it is necessary to increase the number of roots and in special cases like those with low symmetry even to delete Rydberg orbitals from the active space [25,26,250,260]. In the following we will describe briefly the calculations [261]. A detailed report of the vertical excited spectrum of thiophene can be found in references [261,262]. The selection of the active spaces in that work included additional orbitals to minimize the effect of intruder states. The availability of the levelshift technique in later versions of MOLCAS allow us to use a smaller active space.

Config.  Excitation 1  Excitation 2 
VJTU  Inactive (J) Active (V)  Active (U) Active (T) 
VJTIP  Inactive (J) Active (V)  Inactive (I) Active (T) 
VJTIM  Inactive (J) Active (V)  Inactive (I) Active (T) 
ATVX  Active (T) Secondary (A)  Active (X) Active (V) 
AIVX  Inactive (I) Secondary (A)  Active (X) Active (V) 
or:  Active (X) Secondary (A)  Inactive (I) Active (V) 
VJAIP  Inactive (J) Active (V)  Inactive (I) Secondary (A) 
VJAIM  Inactive (J) Active (V)  Inactive (I) Secondary (A) 
BVATP  Active (V) Secondary (B)  Active (T) Secondary (A) 
BVATM  Active (V) Secondary (B)  Active (T) Secondary (A) 
BJATP  Inactive (J) Secondary (B)  Active (T) Secondary (A) 
BJATM  Inactive (J) Secondary (B)  Active (T) Secondary (A) 
BJAIP  Inactive (J) Secondary (B)  Inactive (I) Secondary (A) 
BJAIM  Inactive (J) Secondary (B)  Inactive (I) Secondary (A) 
For more details see Refs. [21,22,265]
The first configuration shown in the thiophene output involves the excitation from the active space to the secondary orbital, which is orbital nr seven of symmetry two (Se2.007). The denominator value for this configuration is close to zero (0.01778941). This is an energy difference, in the approximation. Thus the root state, and some eigenstate of in the interacting space, have almost the same energy value.
Such states, that were not included in the CASSCF configuration interaction but have energies within the range of the lowest CAS states, cause frequent problems in excited state calculations, since they often give small denominators and even, at particular geometries, singularities. We call these states intruders, by analogy to a similar phenomenon in multistate perturbation theory. A calculation of excited states by means of a perturbation theory based on an active space has to deal with the problem of intruder states. This is especially common when large and diffuse basis sets, such as the Rydberg functions, are included in the calculations.
In this example, the coefficient to the first order wave function is large (0.72136094). So is the contribution to the second order energy (0.00509469 H), 0.14 eV. Even worse is the situation for the third term printed involving the fourth orbital (secondary) of symmetry four with an energy contribution of 0.44 eV. The analysis of the secondary orbitals 7 and 4 (they are the first virtual orbital of their symmetry) indicates that they are extremely diffuse orbitals with large Rydberg character. Remember that the subspaces we are using are: frozen (4130), inactive (6040), and active (1503).
This is not the case in the other configurations shown. First we have other ATVX terms including the excitation to the secondary orbital Se2.009. Also we have an AIVX term, involving the excitation from inactive In3.007 to secondary Se3.012. Their contributions to the second order energy, 0.00448260 and 0.00184857, respectively, are not caused by accidental near degeneracies in the value of the denominator. The orbitals involved are not of Rydberg character either. We have finally included as an example the excitation AIVX involving the excitation from In1.010 to Se1.014. Although it has a small value for the denominator, its contribution to the second order energy is very small and therefore it does not represent an important problem.
Intruders can be eliminated by including sufficiently many orbitals in the active space. When this is a reasonable alternative, it is the preferred solution. Limitations in the number of active orbitals can make this approach impractical. However, especially when intruders have clear Rydberg character, their effect on the secondorder energy is often small, except perhaps in a small range of geometries around a singularity due to accidental degeneracy. In this common situation, two other remedies are available: shifting the Hamiltonian, or deleting virtual orbitals. These remedies will be described in some detail in the following.
In order to obtain continuous potential energy functions, one cannot use a casebycase approach, such as deleting an orbital. However, the can be modified in such a way as to eliminate weak singularities. A welltested method is a levelshift technique called LSCASPT2[26,30]. A constant parameter is added to the external part of the zerothorder Hamiltonian. Any denominator close to zero is thus shifted away from zero, and does not produce any singular term. Of course, in a worstcase scenario, it might happen that some other denominator, previously nonzero, is shifted to come close to zero. In general, it is the higher excited states, in combination with large diffuse basis sets and exploration of a large range of geometries, that is the greatest risk for troublesome intruders.
There is also a new, less tried technique, called the imaginary shift method [32]. Here, the use of an imaginary shift value (but taking the real part of the computed correlation energy) offers some advantage, since an imaginary shift cannot introduce new singularities.
With either of the level shift methods, the (2nd order) correlation energy E_{2}
and the (1st order) wave function will depend on
the level shift used. A correction of therefore applied, whereby in practice
this dependence is made small, except of course for the spurious term that has
disappeared. The corrected energy is in fact computed by using Hylleraas' 2ndorder
variational formula to evaluate E_{2}, with the unshifted ,
(10.3) 
To minimize the effect on relative energies, we recommend that the same level shift is used for all states and geometries, if possible. This may require some experimenting. A criterion on absence of disturbing intruders is that the weight of the reference wave function should be roughly the same in all calculations. Without shift, a difference of up to 10% between the weights of the ground and an excited state can be acceptable (that is, the excitation energy is accurate enough) in a CASPT2 calculation without level shift. Using level shift, this should be adjusted to find a better match of reference weights. A detailed explanation of how to use the levelshift technique has been published [31]. Here we will simply summarize the main aspects.
Using the same JOBIPH file as before we perform a new CASPT2 calculation using the input:
&CASPT2 &END
Title
caspt2 input
MultiState
1 6
Shift
0.1
End of input
A levelshift of 0.1 Hartree has been introduced as a separation of the eigenvalues of the zerothorder Hamiltonian. The final energy is then corrected, and the result is:
Reference energy: 551.1062184006
E2 (Nonvariational): .6921992859
Shift correction: .0334372801
E2 (Variational): .7256365659
Total energy: 551.8315878181
Residual norm: .0000003986
Reference weight: .74942
~
CASE SYM ACT IND NONACT INDICES DENOMINATOR RHS value COEFFICIENT CONTRIBUTION
~
ATVX 2 Mu2.0001 Se2.007 .01778941 .00706261 .06072347 .00042887
ATVX 2 Mu2.0001 Se2.009 .20859986 .03118841 .09700134 .00302532
ATVX 4 Mu4.0001 Se4.004 .02156184 .01357269 .11838970 .00160687
AIVX 1 Mu1.0001 In3.007 Se3.012 .28275882 .02231776 .05918658 .00132091
Several details come to our attention. Firstly, the final CASPT2 energy is higher than the result with levelshift 0.0. This is because the introduction of the parameter decreases the amount of dynamical correlation included. Secondly, the weight of the reference function has increased greatly, from 0.29 to 0.74, meaning that the most important intruder states have been removed from the treatment. Finally, we can observe the new contributions of the printed configurations to the second order energy. Configurations involving excitations to the 7 and 4 orbitals have drastically decreased their contributions, proving that the previous contributions were due to degeneracies in the denominators. However, the other two configurations remain almost as they were before, only slightly decreasing their contributions.
Now we use a value for the levelshift parameter of 0.2 Hartree:
Reference energy: 551.1062184006
E2 (Nonvariational): .6619040669
Shift correction: .0557159229
E2 (Variational): .7176199898
Total energy: 551.8235712419
Residual norm: .0000009298
Reference weight: .78212
~
CASE SYM ACT IND NONACT INDICES DENOMINATOR RHS value COEFFICIENT CONTRIBUTION
~
ATVX 2 Mu2.0001 Se2.007 .01778941 .00706261 .03193515 .00022555
ATVX 2 Mu2.0001 Se2.009 .20859986 .03118841 .07304944 .00227830
ATVX 4 Mu4.0001 Se4.004 .02156184 .01357269 .06238180 .00084669
AIVX 1 Mu1.0001 In3.007 Se3.012 .28275882 .02231776 .04673419 .00104300
The observed tendencies are maintained. Finally, a value of 0.3 Hartree:
Reference energy: 551.1062184006
E2 (Nonvariational): .6347955450
Shift correction: .0735679820
E2 (Variational): .7083635270
Total energy: 551.8145819276
Residual norm: .0000006328
Reference weight: .80307
~
CASE SYM ACT IND NONACT INDICES DENOMINATOR RHS value COEFFICIENT CONTRIBUTION
~
ATVX 2 Mu2.0001 Se2.007 .01778941 .00706261 .02173413 .00015350
ATVX 2 Mu2.0001 Se2.009 .20859986 .03118841 .05865340 .00182931
ATVX 4 Mu4.0001 Se4.004 .02156184 .01357269 .04240583 .00057556
AIVX 1 Mu1.0001 In3.007 Se3.012 .28275882 .02231776 .03862959 .00086213
The contributions to the energy are much lower for each increase of the parameter, but we must never forget that we are loosing dynamical correlation with the increase of the levelshift factor. In a calculation of excitation energies that means that the resulting excitation energies become larger each time (dynamical correlation is larger in the excited state). Therefore, the levelshift parameter must be set to the lowest possible value which solves the intruder state problems. In practice it is then convenient to scan all the valence states for several values of the parameter and look for two factors:
We now compute the ground state (GS) also for the levelshift values of 0.1, 0.2, and 0.3, and compare the excitation energies E (always between states computed with the same parameter):
LS (H)  E (eV)  weight GS  weight ES 
0.0  6.11  0.81  0.29 
0.1  6.64  0.82  0.75 
0.2  6.79  0.83  0.78 
0.3  6.89  0.84  0.80 
After checking the remaining states we conclude that a level shift of 0.1 Hartree is enough for our purposes. However the results seem to be too unstable with respect to the increase of the levelshift parameter. As our active space only comprises nine orbitals, we can consider the possibility of increasing it by including two more active orbitals in symmetries and . In this way we minimize the intruder states problems in the best way, by introducing extra (not diffuse hopefully) orbitals. This will increase the accuracy.
The introduction of a (real) levelshift parameter does not automatically remove intruder state problems. It happens that a shift leads to more severe problems that those observed without levelshift. Examples and further explanations are given in e.g. ref. [31]. In such a case is may be possible to find a range of levelshift values where none of the computed states present intruder state problems. In a few cases we have found it necessary to use a shift larger than 0.3 Hartree. Another solution is to try an imaginary shift. This option has not been extensively investigated yet.
Consider a situation like the following:
CASE SYM ACT IND NONACT INDICES DENOMINATOR RHS value COEFFICIENT CONTRIBUTION
~
ATVX 2 Mu2.0001 Se2.004 .30281661 .00194108 .37224517 .00072256
This is a calculation performed using level shift of 0.3 H. (The approximate denominator printed in the listing is that without the added shift). We have added the level shift to solve intruder states problem in other states, but we should use the same technique for all the computed states for consistency reasons (of course always using a ground state computed with the same level shift value). We find, however, that the weight of the CASSCF reference function is lower in the case with level shift 0.3 H (0.61) than in the case without level shift (0.69). In this state we have a denominator with a value close to 0.3 H. As the level shift we apply is a positive quantity (0.3 H) added to this denominator, we have created a problem by decreasing the denominator to a value close to zero. The coefficient of the configuration increases, which is reflected in the contributions to the secondorder energy. Therefore, before applying any level shift, it is wise to check the values of the most important denominators to see if any of them is going to be close to the value of the applied level shift. In those situations we should set the level shift to another value. Sometimes the consequences for the final energy are small (here for instance) but this is not always the case (see ref. [31]).
It is also possible to delete virtual orbitals. This is occasionally used, e.g. when using other types of basis sets than ANO's, in order to delete virtual orbitals that are corecorrelating. The procedure to do that is to take an orbital file, such as that produced by SCF or RASSCF, and edit it by hand and then using it as INPORB file in the RASSCF step. The orbitals one wants to delete are placed at the end of their symmetry group, and the keyword DELEted in used the RASSCF input, indicating how many orbitals are going to be deleted by symmetry. The program will ignore the deleted orbitals, both in RASSCF and the subsequent CASPT2 steps. To obtain accurate energy differences it is necessary to use the same set of initial orbitals and recompute the ground state (or the state one is comparing with) with the same number of deleted orbitals.
When the above scheme is used in order to try to eliminate intruders in CASPT2, the best way is if the INPORB can be prepared from the CASPT2 calculation where the intruder problem occurred.
For that calculation, the natural orbital analysis that follows the CASPT2 calculation shows up a virtual orbital with abnormally large occupation number and diffuse character. Use an editor to move this orbital to the end of the orbital file, and use it as INPORB. When the calculation is repeated, intruders with this orbital heavily populated have been eliminated. Occasionally, several orbitals need to be removed.
The deletion of virtual orbitals works best at singlegeometry calculations, such as obtaining the vertical electronic spectrum.
Let us focus on the MultiState CASPT2 type of calculations. The original reference [15] should be carefully read before using the method. This multidimensional perturbative approach considers the coupling of a number of CASPT2 states, a condition which is crucial to solve certain problems such as adiabatic crossing among states, strong valenceRydberg situations, etc. The treatment is performed for a number of roots of the same symmetry provided they originate from a previous StateAverage CASSCF calculation, that is, the CASPT2 program will use the binary JOBIPH file from a previous SACASSCF calculation, for instance, the six roots ^{1}A_{1} CASSCF calculation in thiophene. The corresponding CASPT2 input to treat simultaneously the six states will be:
&CASPT2 &END
Title
mscaspt2 input
MultiState
6 1 2 3 4 5 6
Shift
0.3
End of input
A level shift parameter of 0.3 au has been selected for comparison with the previous calculations. The program creates a new binary file, JOBMIX, which contains the newly generated Perturbatively Modified (PM) CASSCF wave function.
Using the previous input, the CASPT2 module will perform in a single
run six consecutive singleroot CASPT2 calculations for each one of the CASSCF
states. At the end of each of the calculations the contributions to the Hamiltonian
coupling elements between the computed and the remaining states will be printed.
After computing the six CASPT2 roots, the MSCASPT2 treatment will be performed.
First, the effective Hamiltonian matrix, asymmetric and symmetric, is printed.
Effective Hamiltonian matrix (Symmetric):
1 2 3 4 5
1 .07013926
2 .01263691 .12976380
3 .00071175 .01001560 .18051855
4 .00509735 .00990244 .00321669 .19922802
5 .00607124 .00070650 .00129815 .00225583 .21601193
6 .01998132 .02350235 .00771000 .01037132 .00264941
6
1 .18541807
Notice that the diagonal elements of the matrix correspond to the single root CASPT2 state energies, where some quantity, 551.0 au here, has been added to get a better print of the output. Following, the eigenvalues and eigenvectors of the diagonalized matrix are obtained:
Energies and eigenvectors:
552.07305076 551.88140802 551.81866833 551.80756578 551.79500203
.99308520 .10131857 .01038991 .05207094 .02055799
.07343489 .90295279 .31190606 .28061095 .05245262
.00869768 .19493901 .90626880 .37241673 .03796203
.02478279 .15572120 .13596794 .50373403 .83205915
.02204833 .01553573 .05330075 .08679334 .05789830
.08492920 .33454317 .24485766 .72011863 .54745806
551.78350398
.01655899
.02245882
.02155609
.10285444
.99274682
.05129770
The eigenvalues correspond to the final MSCASPT2 energies, while the eigenvectors describe the combination of the coupled CASPT2 state which give rise to the final MSCASPT2 states. Important: Notice that the states are written in an increasing energy order, and therefore they do not, in general, correspond to the order obtained in the previous SACASSCF calculation. For instance, the MSCASPT2 state number six, energy 551.78350398 au, mainly correspond to the fifth state of the previous calculation. It is very important to remember that the final states are linear combinations of the preceding ones, and therefore a one to one correspondence is hardly possible. In the present example most of the MSCASPT2 states have a strong weight in just one of the preceding states, but this is not the case in many situations. Following in the output, a printing of the new wave function is obtained. It corresponds to linear combinations of the SACASSCF CI wave functions, obtained in the basis of the previous CASSCF averaged orbitals.
The CI coefficients for the MIXED state nr. 1

CI COEFFICIENTS LARGER THAN 0.36
Occupation of active orbitals, and spin coupling
of open shells. (u,d: Spin up or down).
Conf Occupation Coef Weight
11 2 22000 200 .960835 .923204
The CI coefficients for the MIXED state nr. 2

CI COEFFICIENTS LARGER THAN 0.36
Occupation of active orbitals, and spin coupling
of open shells. (u,d: Spin up or down).
Conf Occupation Coef Weight
20 2 2ud00 200 .856751 .734023
The CI coefficients for the MIXED state nr. 3

CI COEFFICIENTS LARGER THAN 0.36
Occupation of active orbitals, and spin coupling
of open shells. (u,d: Spin up or down).
Conf Occupation Coef Weight
85 2 2u0d0 200 .764848 .584993
86 2 2u00d 200 .507350 .257404
The CI coefficients for the MIXED state nr. 4

CI COEFFICIENTS LARGER THAN 0.36
Occupation of active orbitals, and spin coupling
of open shells. (u,d: Spin up or down).
Conf Occupation Coef Weight
1 2 22200 000 .368003 .135427
14 2 22000 u0d .732276 .536229
The CI coefficients for the MIXED state nr. 5

CI COEFFICIENTS LARGER THAN 0.36
Occupation of active orbitals, and spin coupling
of open shells. (u,d: Spin up or down).
Conf Occupation Coef Weight
1 2 22200 000 .416925 .173826
12 2 22000 ud0 .549793 .302272
14 2 22000 u0d .455052 .207072
The CI coefficients for the MIXED state nr. 6

CI COEFFICIENTS LARGER THAN 0.36
Occupation of active orbitals, and spin coupling
of open shells. (u,d: Spin up or down).
Conf Occupation Coef Weight
85 2 2u0d0 200 .517972 .268295
86 2 2u00d 200 .776117 .602358
The comparison of the present wave functions, that will be hereafter called Perturbatively Modified (PM) CASSCF wave functions, and the previous CASSCF wave functions leads to several conclusions. Remember that the orbital basis has not changed, therefore those mixing related to the orbitals are not going to disappear. For instance, state number three will still be formed by two configurations, because the Rydberg 3px character is still delocalized between orbitals 5 and 6 or symmetry . However the character of the second root has changed dramatically. Now one single configuration describes the state, which has acquired a very clear valence character. The previous mixing with a Rydberglike configuration has disappeared. It is illustrative to carry out an additional analysis of the obtained states using the generated file JOBMIX as input file to perform a RASSI calculation, in which new PMCASSCF properties for the states will be obtained. Even when the changes in energies are small, changes in the properties can be considerable. RASSI provides different types of matrix elements (see next section), and dipole moments, transition dipole moments and their directions, and orbital extensions (all of them available from the RASSI output) will be crucial for our purposes in the study of excited states.
Finally, it is necessary to remember that the extent of the MS interaction relies on the mixing of the previous states. This depends on different factors. The basis sets is one of them. The use of one or other atomic basis set to describe the diffuse functions may lead to different answers. It is not uncommon that CASPT2 results with different diffuse basis sets give different answers due to different extents of the valenceRydberg mixing. It will be necessary to perform final MSCASPT2 calculations. Those will change the CASPT2 result in some cases, but it will be unaffected in other cases. Another effect comes from the use of the level shift. The use of MSCASPT2 does not prevent or affect the extent of the intruder effects. Remember that this effect is already included both in the diagonal terms of the effective Hamiltonian as in the nondiagonal coupling terms. Still a careful checking of different LS values and how they affect the CASPT2 values must be performed, and the final MSCASPT2 results should be those in which the effect of the intruder states is small, always trying to use as low level shift values as possible. An alternative is to use an imaginary level shift. Finally, the extent of the offdiagonal coupling elements and its asymmetric character introduce further inaccuracies in the treatment. In most cases the proper enlargement of the active space diminishes most of the spurious effects and increases the accuracy.
One powerful tool included in the MOLCAS package is the RASSI program. RASSI (RAS State Interaction) forms matrix elements of the Hamiltonian and other operators in a wave function basis which consists of individually optimized CI expansions from the RASSCF program. It also solves the SchrÃ¶dinger equation within the space of these wave functions. In spectroscopy we need to compute the matrix elements of a oneelectron operator such as the dipole transition moment to obtain the intensity of the transitions. In an absorption process this means computing the interaction of the ground state with the excited states. RASSI will compute all matrix elements among the states provided they have been computed with the number of inactive and active orbitals, and using the same basis set. The transition dipole moments are computed using the length representation.
In our example we have used two different active spaces. We therefore need to perform at least two RASSI calculations. First we will compute the interaction of the ground state 1^{1}A_{1} (computed as single root), with the ^{1}A_{1} and ^{1}B_{2} excited states. We should link the corresponding JOBIPH files:
ln fs $Project.11A1.JobIph JOB001
ln fs $Project.1A1.JobIph JOB002
ln fs $Project.1B2.JobIph JOB003
and use the RASSI input file:
&RASSI &END
Nrofjobiphs
3 1 5 5
1
2 3 4 5 6
1 2 3 4 5
End of input
As we are using states that are not orthogonal (this is the case among the 1^{1}A_{1} ground state computed as a single root and the other ^{1}A_{1} states) we must take the matrix elements of the transition dipole moment computed after the transformation to the eigenbasis; the second time they appear in the output:
PROPERTY: MLTPL 1 COMPONENT: 2
ORIGIN : .00000000D+00 .00000000D+00 .00000000D+00
STATE : 1 2 3 4
~
1 .00000000D+00 .00000000D+00 .43587844D+00 .00000000D+00
2 .00000000D+00 .00000000D+00 .10019699D+01 .00000000D+00
3 .43587844D+00 .10019699D+01 .00000000D+00 .46859879D+00
4 .00000000D+00 .00000000D+00 .46859879D+00 .00000000D+00
5 .90773544D01 .75718497D01 .00000000D+00 .27645327D+00
6 .00000000D+00 .00000000D+00 .41227462D+01 .00000000D+00
7 .00000000D+00 .00000000D+00 .89741299D+00 .00000000D+00
8 .16935368D+00 .15487793D+01 .00000000D+00 .41013917D+01
9 .81381108D+00 .79559359D+00 .00000000D+00 .88184724D01
10 .00000000D+00 .00000000D+00 .43659784D+00 .00000000D+00
11 .13520301D+01 .50454715D+00 .00000000D+00 .56986607D01
~
...
~
PROPERTY: MLTPL 1 COMPONENT: 3
ORIGIN : .00000000D+00 .00000000D+00 .22419033D+01
STATE : 1 2 3 4
1 .28126942D+00 .92709234D+00 .00000000D+00 .11876829D+00
2 .92709234D+00 .26218513D+00 .00000000D+00 .14100968D+00
3 .00000000D+00 .00000000D+00 .52558493D01 .00000000D+00
4 .11876829D+00 .14100968D+00 .00000000D+00 .36996295D+00
5 .00000000D+00 .00000000D+00 .43197968D+01 .00000000D+00
6 .15470487D+00 .42660550D+00 .00000000D+00 .94593876D+00
7 .18676753D01 .18738780D+01 .00000000D+00 .37737952D+01
8 .00000000D+00 .00000000D+00 .28182178D+00 .00000000D+00
9 .00000000D+00 .00000000D+00 .38253559D+00 .00000000D+00
10 .12859613D+01 .48476356D+00 .00000000D+00 .35525361D+00
11 .00000000D+00 .00000000D+00 .39325294D01 .00000000D+00
We have a symmetric matrix containing the results. The matrix elements corresponding to the interaction of the first state in the input (ground state) and the remaining states appear both in the first column and in the first row (only partially printed here). Remember that the transition dipole moment (TDM) matrix elements are determined by the symmetry. The matrix element will be zero for the x and y components of TDM, and nonzero otherwise. The matrix element will be nonzero only for the y component of TDM. This is because the product (wave function 1 x dipole moment component x wave function 2), if decomposed into irreducible representations, must contain the totally symmetric representation to have an allowed transition. In this simple case, we can use a multiplication table for the irreps. Thus, for instance, ( ^{1}A_{1} (z) x TDM_{y} x ^{1}A_{1} (z) ) gives y, which does not belong to the totally symmetric representation. A look at the character table and the behavior of the x,y,z functions will give us the information we need.
Therefore, in the component two (y) of the transition dipole moment matrix elements we have zero values for the interaction among ^{1}A_{1} states and nonzero values for the interaction among ^{1}A_{1} and ^{1}B_{2} states.
The RASSI program in 6.0 and later versions of MOLCAS will print the oscillator strengths and the Einstein A coefficients for all transitions. Also the angles of the transition moment vectors to the coordinate axes will be printed. In the calculation RASSI will use the energies given as input, so be careful to use the keywords HDIAG or EJOB to use energies which include dynamic correlation.
We illustrate how the oscillator strengths are computed. The 11 states are ordered by CASSCF energies. We focus on the valence states; firstly the fourth and fifth ^{1}B_{2} states. Their transition dipole moment values in atomic units are 0.81381108 and 0.13520301D+01, respectively. The oscillator strength is defined as:
The energy difference E is the excitation energy expressed in atomic units. The transition moments were computed by CASSCF. It is usually not practically possible to compute them with dynamic correlation included, except if a common set of orbitals are used. However, the CASSCF values are usually good enough. (Exceptions occur, e.g. close to narrowly avoided crossings or conical intersections). The excitation energies, on the other hand, are quite sensitive to dynamic correlation. Thus, it is a good approach to use CASSCF TDMs and CASPT2 excitation energies. The values for the oscillator strengths of the two ^{1}B_{2} valence states are 0.086 and 0.324, respectively. The excitation energies are 5.31 and 7.23 eV, respectively. All data corresponds to results obtained using the 0.1 Hartree value for the levelshift parameter.
Remember that in other symmetries like C_{2h} the ^{1}B_{2} states have two components of TDM, x and y, for which the matrix elements with respect to the ground state are nonzero. In this case the TDM^{2} value is computed as TDM_{x}^{2} + TMD_{y}^{2}. In those cases is is also possible to compute the direction of the total TDM vector by taking their components and compute the angle respect to any of the axis.
You will find the complete calculation of the absorption spectrum of thiophene in reference [249]. You can observe that, despite there being no levelshift technique used, the final results on the excitation energies agree to within 0.1 eV to those shown here.
Thiophene has a valence orbital space small enough to allow the simultaneous inclusion of all the corresponding Rydberg orbitals into the active space (remember valence space (1302) + Rydberg spaces (0201) or (4020)), but this is not always the case. In addition, the valenceRydberg mixing is not severe. This mixing is reflected in the orbital extension or the population analysis. In difficult cases valence and Rydberg orbitals mix, and then the configurations also mix. Valence states become more diffuse and Rydberg states more compact. Energetically this has minor consequences for the Rydberg states, which can be computed using these CASSCF mixed wave functions. This is not the case for the valence states. They are extremely sensitive to the mixing. Therefore, if we do not observe clear and compact valence states some mixing has occurred.
We consider the example of the guanine molecule, the nucleic acid base monomer. It is a system with 11 valence orbitals which should be included into the active space. It is a planar system in the C_{s} point group. Focusing only in the states we can label the active orbital space (0,11) where 0 is the number of a' orbitals and 11 the number of a'' orbitals. In C_{s} symmetry the Rydberg orbitals are distributed as (6,3), using the same labeling. Therefore the calculation of the corresponding A' states should use the space (0,14) with 14 active electrons and a large number of roots. This is a large calculation that one might want to avoid. One can perform several test calculations (maybe even RASSCF calculations) and find if any orbitals can be excluded. The lowest occupied orbital is a deep orbital which does not participate in the lowest valence excited states and can be excluded from the active space. Despite this exclusion, a (0,13) orbitals calculation is still expensive. We can proceed in another way. Consider the new valence space (0,10), and add only one more orbital designed to include the first Rydberg orbital. With this space of (0,11) orbitals and 12 active electrons we perform a CASSCF including 6 roots.
Our basis set is of the ANOL type contracted to C,N,O 4s3p1d / H 2s, plus 1s1p1d optimized diffuse functions placed in the cation charge centroid. The results are collected in Table .
State  Theoretical  Experiment^{b}  
CAS  PT2  f  f  
– transitions  
2^{1}A'  5.72  4.47  .20  64^{o}  1.07  4.44.5  .16  (4^{o},35^{o})  
3^{1}A'  6.74  5.30  .09  +52^{o}  2.72  4.95.0  .25  (75^{o})  
4^{1}A'  7.18  5.63  .05  90^{o}  3.10  5.75.8  <.05^{c}  
5^{1}A'  8.45  6.83  .26  0^{o}  3.20  6.16.3  .41  (71^{o},79^{o})  
^{a}See ref. [266] for details. 
There are important discrepancies between theoretical and experimental results, more important in the properties such as the intensities and the transition dipole moments than in the excitation energies. If we analyze the CASSCF output everything is apparently correct: six converged roots, all of them clear valence states, and no Rydberg orbital into the active space. This is the problem. At least one of the Rydberg orbitals should have been introduced into the active space. Rydberg and valence orbitals must be treated simultaneously and this is not possible if there is no Rydberg orbital in the active space.
The correct way to proceed is to take the first Rydberg orbital (3p_{z}) and place it as the 11th active orbital of a'' symmetry. Then the CASSCF calculation will retain it in the space. Once the calculation has converged we observe than at least one of the computed states is of Rydberg character. It can also happen that some mixing appears in the valence states due to the presence of the diffuse orbital in the active space. The Rydberg orbital is then removed (placed in the last position of its symmetry and the DELEte option used) from the active space and the calculation repeated. This time the next Rydberg orbital (3d_{xz} or 3d_{yz}) will take its place. The process is repeated once again until the three Rydberg orbitals have been first included in the active space and then deleted (option DELEted of the RASSCF program). Now we can reduce the active space to (0,10), only including valence orbitals and valence excited states.
We can repeat the calculation including even more roots. The results are in Table .
=0pt
=0pt
State  Theoretical  Experiment  
CAS  PT2  f  f  
– transitions  
2^{1}A'  6.08  4.76  .133  15^{o}  7.72  4.44.5  .16  (4^{o},35^{o})  
3^{1}A'  6.99  5.09  .231  +73^{o}  6.03  4.95.0  .25  (75^{o})  
4^{1}A'  7.89  5.96  .023  +7^{o}  5.54  5.75.8  <.05^{c}  
5^{1}A'  8.60  6.65  .161  80^{o}  10.17  6.16.3  .41  (71^{o},79^{o})  
6^{1}A'  9.76  6.55  .225  41^{o}  6.11  
7^{1}A'  8.69  6.66  .479  +43^{o}  6.57  6.66.7  .48  (9^{o},41^{o})  
8^{1}A'  9.43  6.77  .098  +52^{o}  7.17  
^{a}See ref. [266] for details.  
^{b}A better match with the experimental values is obtained by considering solvent effects. 
The experience of this type of treatment in different molecules [26,30,266] points out that if the valence states of a molecule are computed without considering the Rydberg states and functions (whether by excluding them from the basis set or from the active space) can result in an additional CASPT2 error as large as 0.30.4 eV. The errors are more severe for other transitions properties. One example of this can be found for two different CASPT2 treatments of the formamide molecule, one including diffuse functions and other excluding them (see ref. [267] for details). Notice, however, that this approach cannot describe a true valenceRydberg mixing. An alternative to such an approach is to use the MultiState CASPT2 treatment that, although computationally expensive, might properly treats the valenceRydberg mixing. It must be remembered, however, that the performance of the MSCASPT2 method relies on the previous mixing of the wave functions, and therefore it will not be unusual, depending on the employed basis set, to obtain CASPT2 results that already give the same answer as MSCASPT2 results when the initial basis sets are changed.
The calculations become increasingly difficult with increased size of the system or in low symmetry cases. Common problems one has to solve are the selection of the active space when it is not possible to include all orbitals expected to be important and the presence of artificial valenceRydberg mixing in the description of the states. Specific problems appear in systems containing transition metals, where there are a large amount of states close in energy.
To include all the required orbitals into the active space is sometimes impossible. This is one of the important limitations of the methodology. But some solutions are available if one is aware of the limitations. References [268] and [269] report studies on the porphin and indigo molecules, respectively. Porphin and indigo have 24 and 20 orbitals, respectively. It is obviously impossible to include all of them in the active spaces. The analysis of the configurations and occupation numbers of the orbitals in a restricted number of excited states by means of the RASSCF method has been found to be a useful procedure to find a proper active space to study different states of the systems. The RASSCF method is able to deal with a larger number of configurations making possible to include all the orbitals in the active space and analyze the role of the different orbitals. Our goal in this case is to be able to discard some of the deepest or highest orbitals if they become less important in the description of the desired states.
One possibility is to perform a SDTQ calculation involving all the presumably important active space (occupied orbitals in RAS1, empty orbitals in RAS3, no orbitals in RAS2, and four holes/electrons allowed in RAS1/RAS3). The occupation numbers for the active orbitals obtained for such calculation are usually similar to those of a full CASSCF treatment. Another possibility is to place in the CAS space (RAS2) the most important orbitals and the corresponding electrons and only allow singles and doubles excitations from RAS1 (occupied orbitals) to RAS3 (empty orbitals). In all these cases we will study the configurations and occupation numbers of the orbitals to find if some of them are or minor importance for the description of the states we are considering and then reduce the active space for the CASSCF/CASPT2 calculation [268,269].
Calculation on the excited states of transition metal compounds have to deal with another set of problems. For instance, the known 3d doubleshell effect: two sets of d orbitals (3d and 4d) must be included in the reference space in order to obtain accurate results [26] in molecules containing metal atoms of the first transition row with many delectrons (FeZn). This is a severe limitation when more ligands are included together with the metal atom. Illustrations of such problems are the calculation of the cyanide and carbonyl transition metal compounds [26,270] and metalprotein models [271]. Corevalence [272] and relativistic effects [31] have been shown to be important for obtaining accurate results. Finally, the problem of the high multiplicity states in the standard CASPT2 formulation has to be considered. The zerothorder Hamiltonian is defined as a Focktype oneelectron operator. Apart from the originally proposed Fock matrix [21,22], a correction, denoted g_{1} [29], has been designed so that CASSCF wave functions dominated by a closedshell configuration, on the one hand, and an openshell configuration, on the other hand, are treated in similar and balanced ways in the perturbation calculation. This correction was shown to be essential in order to obtain reliable results for the Cr_{2} molecule with the CASSCF/CASPT2 method [30].
Each type of system and situation has its own specific problems. Size and convergence problems in systems without any symmetry [273,274], symmetry breaking and localization problems in high symmetry cases [275], excited states in radical cations [276] and anions [277], etc. In addition, there are situations such as the crossing regions which require the simultaneous treatment of more than one state at the CASPT2 level, which can only be solved using the multistate option in CASPT2.