CASSCF and NEVPT2 calculations: Ground and excited states of multireference systems. A case study of Ni(CO)4 and the magnetic system NArO

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1 CASSCF and NEVPT2 calculations: Ground and excited states of multireference systems. A case study of Ni(CO)4 and the magnetic system NArO The ground states of many molecules are often well described by a single determinant. In such cases it is often most appropriate to use so-called single reference methods. Density functional methods (DFT) are widely used, and excited states can be described by the corresponding time-depent DFT (TDDFT approaches). Alternatively one can start from Hartree-Fock (HF), and apply coupled cluster methods. For ground states the CCSD(T) aproach is most suitable, and in ORCA one has very efficient local CC approaches that go by the name DLPNO-CCSD(T). For excited states one can use EOM-CC or STEOM-CC. In ORCA, one can run efficient DLPNO-STEOM or bt-steom calculations, that employ the efficient DLPNO technology. These techniques are described in other labs. Here I will focus on systems or states that require a different (more complicated) treatment. They are characterized by a (small) set of active orbitals, and wave functions that are qualitatively well described by determinants that vary in the occupation of these active orbitals. They can have additional (deeper lying) orbitals that are all doubly occupied. Such cases often arise if we are intersted in both ground and excited states that can both be described by active space configurations. Examples of such systems are biradicals (e.g benzene with two H-atoms removed), open-shell atoms, magnetic systems (essentiall open-shell atoms connected by bridging ligands), and many systems involving transition metal atoms. The latter systems often have a number of close lying orbitals comprised of the d orbitals of the metal. The same can happen for f-shell atoms (actanides or lathanides). The calculation proceeds in a number of steps. The first step is the selection of a space of active orbitals, and optimization of the orbitals. This is done using a complete active space CASSCF calculation. This type of calculation can provide qualitatively accurate wavefunctions, and geometries, similar to HF for single reference molecules. However, to get more accurate energetics and excitation energies one should include additional electron correlation effects. In ORCA the NEVPT2 approach is an efficient and accurate way to do this. In the Nooijen lab we have developed the MREOM Coupled cluster approach. This is more expensive, but is expected to be more accurate than the perturbative NEVPT2 approach. This will be discussed in other labs. Other MRCI and DDCI like approaches are available in ORCA. However to pursue MRCI calculation the MOLPRO package may be more suitable. Here I will focus on the CASSCF and NEVPT2 approach and some of their variations. Detailed Case study Ni(CO)4 I will treat one example in detail, the tetrahedral Ni(CO)4 molecule. This molecule has a high symmetry, and we would like to preserve the degeneracy patterns of the true energy levels. The first step in the calculation is the creation of a suitable CASSCF calculation to calculate a number of low-lying electronic states using one type of so-called state-averaged calculation. All of the calculations can be found in the NEVPT2 directory on chem400a/b.

2 The calculations discussed in this section are all in NEVPT2/NiCO4/10e_8o. I will also briefly discuss a different type of CAS, using 10 electrons in 10 (spatial) orbitals. A good starting point to get an idea of the orbitals is often a proceeding DFT calculation. I started with a fixed geometry (that I obtained from elsewhere), and run the following DFT bp86 calculation, asking to print the orbitals (normalprint): Input file bp86_orb.inp! bp86 def2-svp NormalPrint * xyz 0 1 Ni C C C C O O O O * This yields the following orbital energies around the Fermi level: We notice a nice degeneray pattern showing the E and T representations which are 2- and 3-fold degenerate respectively. The 5 occupied orbitals (2.0000) are d-orbitals on Ni, bonded to the carbonyl COs. Likewise the first five orbitals above the fermi level (0.0000) have significant d-type character. This can be seen by inspecting the orbital character in this section of the output: LOEWDIN REDUCED ORBITAL POPULATIONS PER MO

3 THRESHOLD FOR PRINTING IS 0.1% If we scroll down towards the fermi level we see: Ni pz Ni px Ni py Ni dz Ni dxz Ni dyz Ni dx2y Ni dxy C s C pz C px Showing the predominant d-character of the last 5 orbitals (37-41). Likewise if we continue and examine the virtual orbitals: we see some d-character in orbitals 42-46, but not in other low-lying virtuals, e.g Orbital 50 is the Ni 4s orbital: Ni pz Ni px Ni py Ni dz Ni dxz Ni dyz Ni dx2y Ni dxy This suggests we can use an active space with 10 orbitals (5 occ + 5 vrt), and 10 electrons. Let us now do an initial CASSCF calculation, optimizing just one singlet state Input file CAS_a.inp!CASSCF def2-svp norb 10 nroots 1

4 We read in the orbitals from the previous pb86 calculation ( section), and calculate the ground state CAS. I am doing this all with a small basis set, as we do this to learn the procedure. I would prefer to use the def2-tzvp basis set in general for such calculations. Even then it makes sense to first explore things with a smaller basis set first. We can examine the relevant parts of the output. Look for the statement Final CASSCF energies, Final CASSCF energy : Eh ev ORBITAL ENERGIES NO OCC E(Eh) E(eV) Scrolling down, to were the occupation numbers are partial (no longer 2.000): We will be interested in a number of low-lying singlet states, and we need to figure out how many, and what the size of the final active space will be. To this we use the orbitals from the previous CAS_a calculation to start from, and ask for many states in the singlet manifold. However we do not optimize the orbitals (NoIter). Here is the input file Input File CASCI_a.inp!CASSCF def2-svp NoIter

5 norb 10 nroots 20 We asked for 20 roots here. To examine the relevant part of the output search for Final CASSCF energy, and then look for (or scroll down to) CAS-SCF STATES FOR BLOCK 1 MULT= 1 NROOTS= ROOT 0: E= Eh [ 0]: [ 2916]: [ 1060]: [ 841]: [ 168]: [ 27]: [ 217]: [ 2965]: [ 3184]: [ 792]: [ 76]: [ 436]: [ 6]: [ 3936]: [ 1812]: [ 6060]: ROOT 1: E= Eh ev cm** [ 2907]: [ 785]: [ 2]: [ 21]: This lists a number of excitation energies. We see nice degeneracy patterns, 3 states at ev, 3 at , 2 at , and so on. These energies are very high. Let us make a second attempt, by only asking for 4 roots in a CASSCF calculation (optimize orbitals).!casscf def2-svp norb 10

6 nroots 4 We can run the CASCI_a2 calculation afterwards, asking to calculate 20 roots, but now with the optimized orbtals from the above 4-state CAS(10e,10o). Looking for the converged energies we now find the excitation energy pattern: 1* ev, 3*6.028 ev, 2*6.169, 3*6.475, 3* Then there is a jump to ev This indicates we might look at 12 low-lying states. Moreover looking at the occupation numbers before final CASSCF energy, we see N(occ)= This indicates that the last two orbitals are not strongly occupied, and we might reduce the CAS to 12 states, 10e, 8o This leads to the following CAS_b input file:!casscf def2-svp norb 8 nroots 12 Which has final occupations: N(occ)= You can also find the final state energies, 0.000, 4.938, and so on. We can observe that the degeneracy pattern is not perfect. In particular the last ones ( etc.). The reson is that the CASSCF is not sufficiently converged. As long as we are stll exploring this makes perfect sense. However, once we are happy with the CAS it may make sense to converge energies more sharply (ExtermeSCF), as in CAS_c.inp!CASSCF def2-svp ExtremeSCF

7 norb 8 nroots 12 You can check occupation numbers and excitation energies to see that degeneracy patterns are nearly perfect now which shows perfect degeneracies. You can also examine the energies of the 12 states (just above Final CASSCF energy): CI-ITERATION 16: ( 0.03) CI-PROBLEM SOLVED DENSITIES MADE E(CAS)= Eh DE= Energy gap subspaces: Ext-Act = Act-Int = N(occ)= g = Max(G)= Rot=46,43 We have found the CASSCF solution we want. Now it is strightforward to run NEVPT2 calculations: Input file NEVPT2.inp!CASSCF def2-svp NoIter

8 norb 8 nroots 12 NEVPT2 SC We read in the orbitals from the previous CAS_c calculation, and can use NoIter therefore. The NEVPT2 keyword (lots of documentation in the ORCA manual) can be included in the CASSCF input section. NEVPT2 SC is the fastest version of the NEVPT2 variants. This yields the following output in the NEVPT2 section: NEVPT2 TRANSITION ENERGIES LOWEST ROOT (ROOT 0, MULT 1) = Eh ev STATE ROOT MULT DE/a.u. DE/eV DE/cm**-1 1: : : : : : : : : : : You can also look at the absorbtion (or CD) spectrum. At thi spoint we can examine some other variations of NEVPT2 calculations. The first is a potentially more accurate variant, quasi-degenerate NEVPT2, using the input file qd_nevpt2.inp!casscf def2-svp NoIter

9 norb 8 nroots 12 NEVPT2 SC NEVPT qdtype 1 which yields both the original NEVPT2 and the corrected QD-NEVPT2 energies: NEVPT2 TRANSITION ENERGIES LOWEST ROOT (ROOT 0, MULT 1) = Eh ev STATE ROOT MULT DE/a.u. DE/eV DE/cm**-1 1: : : : : : : : : : : QD-NEVPT2 TRANSITION ENERGIES LOWEST ROOT (ROOT 0, MULT 1) = Eh ev STATE ROOT MULT DE/a.u. DE/eV DE/cm**-1 1: : : : : : : : : : :

10 A final possibly very useful option is to use dlpno_nevpt2. This can be a bit expensive for small molecules, but it is much more efficient when you get to really large molecules. Here is the input file:!dlpno-nevpt2 def2-svp def2-svp/c Nofrozencore norb 8 nroots 12 You need to specify a fitting basis set (def2_svp/c here), and you can set the method on the first input line. The calculation takes more than an hour. The efficieny kicks in for larger molecules. Nofrozencore is recommed for all NEVPT2 calculations. More Examples. For the Ni(CO)4 molecule the choice of CAS was not so clear. We decided to do 12 states in a CAS(10e, 8o), but a clear alternative was to use 12 states with (10e,10o). These calculations are also done in the NEVPT2/NiCO4/10e_10o directory. You can examine the differences. This shows such calculations can be delicate. Finally the calculations may be suitable for magnetic systems. The NEVPT2 calculations will run the fastest among available multireference methods. I provided results for a simple NArO system. This is considered in more detail in the MREOM write-up and directory. The NEVPT2 results compare fairly well with MREOM energies and they are often an enormous improvement over CASSCF energies themselves. The DDCI approach is also a well-established approach for magnetic systems, and inclusion of spin orbit effects are straigtforward. It is possible to use Spin-Orbit Coupling with NEVPT2, but this needs a detailed investigation. I did provide in an input file avg_nevpt2_soc, which uses the avg_nevpt2 energies in a CASCI_soc calculation, as discussed in the SOC section of the ORCA manual. I did not make a detailed write-up for this system. You can follow the input and output files in the NEVPT2/NArO directory, as well as the joball file (which has comments), which reveals the full history.

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