MRCI calculations in MOLPRO

Transcription

1 1 MRCI calculations in MOLPRO Molpro is a software package written in Fortran and maintained by H.J. Werner and P.J. Knowles. It is often used for performing sophisticated electronic structure calculations, in particular of multireference CI type and also CCSD(T) calculations. The program is also suitable to do ab initio calculations on very large molecules using local correlation techniques, and to include an explicit r12 factor in the wave function. The latter option allows one to get very accurate results using relatively small basis sets. Nowadays Molpro benefits from the contributions of a number of authors. Moreover, this program is updated often and the complete software package includes many of the most current techniques used to solve electron structure problems. Since Molpro is quite sophisticated, the input files for the various exercises are provided to you. Your objective is to follow through the details provided here as a way of familiarizing yourself with the structure of the input and output files of Molpro. The input files for this lab are located in ~nooijen/chem440/molpro 1. Calculating the potential energy surface for the N 2 molecule (directory N2_pes). Let us start with simple single point energy calculations. The input file n2_re.inp provides the molpro input to run this calculation. To run the calculations we use a similar protocol as when running ACES2 calculations. Use a submit file in your working directory, and use a joball file to specify the calculations you wish to run. Here the joball file might read jobmolpro n2_re As with ACES2 the script jobmolpro assumes the extension for the input file (.inp) while it writes the output file with extension (.out), as it is running, and it changes it to (.out0) when the calculation is finished. This allows you to keep track of calculations. The input file for the single point calculation reads explicitly: ***,N2 Single point symmetry,x,y,z!use D2h symmetry!z-matrix N,N,Re basis=6-31g*!define basis set Re=2.1!define r for this geometry hf;!do Hartree-Fock SCF calculation escf=energy!save scf energy for this geometry ccsd(t);! run CCSD(T) calculation eccsd_t=energy! save ccsd(t) energy for printing eccsd=energc! energyc is the ccsd energy in Molpro casscf;!do CASSCF calculation. Use default active space ecasscf=energy!save CASSCF energy mrci;!do MRCI calculation. Use defaults for the orbital spaces emrci=energy(1)!save MRCI energy for first state emrci_q=energd0(1)!save MRCI+Q energy for first state (Davidson correction). {table,re,escf,ecasscf,emrci,emrci_q,eccsd,eccsd_t,!produce a table with results head, Re,HF-SCF,CASSCF,MRCI,MRCI+Q,CCSD,CCSD(T)!Define column headers for table title,results for N2, basis $basis!title for table sort,1,2,3!sort table (if neceessary) 1

2 2 The input contains comments to explain what is done (everything on a line following the exclamation mark is comment). Here we run a Hartree-Fock calculation followed by a CCSD(T) calculation, a CASSCF calculation followed by a MRCI calculation. All of the relevant energies are assigned to variables that are then printed in a Table. You can go through the Molpro output, which is very compact. At the end of the file, you find the table we asked the program to print. Results for N2, basis 6-31G* RE HF-SCF CASSCF MRCI MRCI+Q CCSD CCSD(T) We used a fairly simple input here to run all of the above calculations. Molpro chooses its defaults very carefully, and I myself find this quite impressive. It is not an easy task to figure this out in an automated fashion. Let us next calculate the potential energy surface for the N 2 molecule, using the CCSD(T) method. This method is very accurate around the equilibrium bond distance, but it breaks down at larger distances. If you try to run CCSD(T) beyond 4.2 Bohr, the calculation fails to converge. Even at 4.0 Bohr the result is no longer accurate. In Molpro it is very easy to calculate a potential energy surface. We can simply provide a list of distances. Here is the input (file n2_cc_pes.inp): ***,N2 potential symmetry,x,y,z!use D2h symmetry!z-matrix N,N,r(i) basis=6-31g*!define basis set distances=[1.5,1.6,1.7,1.8,1.9,2.0,2.1,2.2,2.3,2.4,2.5,2.6,2.8,3.0,3.2,3.4,3.6,3.8,3.9,4.0]!list of distances do i=1,#distances!loop over distances for N-N (in Bohr or atomic units) r(i)=distances(i)!save r for this geometry hf;!do Hartree-Fock SCF calculation escf(i)=energy!save scf energy for this geometry ccsd(t),maxit=100,nocheck;!run CCSD(T) calculation, with some further options. eccsd_t(i)=energy!assign ccsd(t) energy to array eccsd(i)=energc! assign ccsd energy to array enddo!end of do loop i {table,r,escf,eccsd,eccsd_t!produce a table with results head, R,HF-SCF,CCSD,CCSD(T)!modify column headers for table save,n2_cc_pes.tab!save the table in file n2_cc_pes.tab title, Potential energy surface for N2, basis $basis!title for table sort,1,2,3!sort table In setting up the calculation you are creating simple loop structures. Also if statements are possible, inside the input file! Here, we loop over the distances and at each distance, i.e. geometry r(i) we run an scf and CCSD(T) calculation. The resulting energies are saved in arrays that we can name on the fly. At the end of the input file we collect the data in a table. This table is saved as n2_cc_pes.tab and copied back to your working directory. This is what it looks like (a piece of it): 2

3 3 Potential energy surface for N2, basis 6-31G* R HF-SCF CCSD CCSD(T) Likewise we can run MRCI calculations. The input file is in n2_mrci_pes.inp, and is replicated here: ***,N2 MRCI potential symmetry,x,y,z!use D2h symmetry!z-matrix N,N,r(i) basis=6-31g*!define basis set distances=[1.5,1.6,1.7,1.8,1.9,2.0,2.1,2.2,2.3,2.4,2.5,2.6,2.8,3.0,3.2,3.4,3.6,3.8,3.9,4.0,4.2,4.6,5.0] st of distances do i=1,#distances!loop over distances for N-N (in Bohr or atomic units) r(i)=distances(i)!save r for this geometry hf;!do Hartree-Fock SCF calculation escf(i)=energy!save scf energy for this geometry casscf;!do CASSCF calculation. Use default active space ecasscf(i)=energy!save CASSCF energy mrci;!do MRCI calculation. Use defaults for the orbital spaces emrci(i)=energy(1)!save MRCI energy for first state emrci_q(i)=energd0(1)!save MRCI+Q energy for first state (Davidson correction). enddo!end of do loop i {table,r,escf,ecasscf,emrci,emrci_q!produce a table with results head, R,HF-SCF,CASSCF,MRCI,MRCI+Q!modify column headers for table save,n2_mrci_pes.tab!save the table in file n2.tab title, Potential energy surface for N2, basis $basis!title for table sort,1,2,3!sort table!li\ In the table n2_mrci_pes.tab (see where we specified this name in the input file), we collected the results at the HF-SCF, CASSCF, MRCI and MRCI+Q level. In general one would expect the MRCI+Q calculations to be most accurate. In our trial calculations we use a poor basis set, 6-31G*. To get accurate results a bigger basis set is needed, for example CC-PVTZ. This is very easy to do. Just change the basis set in each of your inputs and rerun the calculations. I have collected results in the N2_pes_TZ directory. The tables can directly be imported in excel, and you can plot the potential energy surfaces. They could be used to calculate accurate vibrational and rovibrational energies, or example using the LEVEL program developed by Professor Bob Leroy. You need to ask him for advice on how to use it! 2. Calculating electronic excitation energies of atoms: Nitrogen Atoms are well suited for testing the validity of electronic structure calculations because their wavefunctions have no contributions from rotational or vibrational motion. The hydrogen atom is the simplest atom and the solutions of the Schrödinger equation for 3

4 4 the H atom are taught in any introductory quantum mechanics class. For many-electron atoms exact solutions are not known, and an accurate technique to calculate these socalled multireference systems is to use the multireference CI method. Many-electron atoms are quite complicated because of the degeneracy of their energy levels. Many of these levels cannot be described using Hartree-Fock MO theory, but require the use of CASSCF and subsequent MRCI treatment. We will use the atomic calculations to explain more about MRCI and the various orbital spaces. As molecules (or atoms) get bigger one would need to properly select the input parameters. Let me first quickly recall what we learn on atoms in a first course on quantum mechanics. The electronic configuration of Nitrogen is 1s 2s 2p. This means we have 3 electrons in 6 spin-orbitals of p-type, which means one can distribute them in 6 6! = = = 20 ways, or we have !3! micro states. The state of highest MS = S =, ML = L= 0. This is the 4 S multiplet, 2 which accounts for 4 states in total. Any multiplet is always (2S+1)*(2L+1)-fold degenerate. Creating the state of highest M L = = 2 gives rise to the 2 D state, with 1 M =±. There are 6 more states that lead to a 2 P multiplet. In Molpro we would S 2 calculate only the state of the highest possible M S for each multiplet. This means we should be able to get 1 state of the 4 S multiplet, 5 states representing the 2 D and 3 states for the 2 P. Let us see how to accomplish this in Molpro. In order to specify the state we can use both symmetry and spin-multiplicity. Molpro uses the quantity 2S to specify spin-multiplicity, which is always an integer. To indicate the spatial symmetry we use the D 2h abelian subgroup of the full SO(3) point group. From the table below we can glean the symmetries resulting from occupying different orbitals. The so-called highest Abelian pointgroup symmetry of the nitrogen atom is D 2h. It has the following irreducible representations. Number Name Function 1 A g s 2 B 3u y 3 B 2u z 4 B 1g xy 5 B 1u z 6 B 2g xz 7 B 3g yz 8 A u xyz It is not so easy to explain how to use this information to determine the symmetry labels of atomic multiplets. Let me give you the results and some hand waving arguments. 4

5 5 For the 4 S we occupy pxpp y z. This transforms like xyz which corresponds to A u in symmetry block 8. This state preserves the spherical symmetry, and the degeneracy of the orbitals. The half-filled state is a nice and easy state to calculate. For the 2 P the states transform in the same way as the px, py, p z orbitals themselves, or like xyz,,. This corresponds to representations B1 u, B2u, B 3u in symmetry blocks 2, 3, 5. Corresponding determinants look like pxpxpz + pypypz, which transforms like z. The 2 D state is most complicated to explain. The xy, xz, yz components correspond to blocks 4, 6, and 7. There are two more states x y, z. They are in symmetry block 1. However, each D-state has three p-orbitals occupied, not 2. I have determined the symmetry block of the state, but pretending to look at the d-orbitals from the atom. In fact this is sufficiently correct. The symmetry (using the abelian subgroup) works out to be the same. We will get back to this later on. Let us now try the Molpro calculations. We will first calculate the 4 S state using CCSD(T). The calculation of the 4 S is easiest. It has an spin up electron in each of the 3 p-orbitals. We need to specify an unrestricted CCSD(T) calculation as the atom is openshell. The input files in this section are all in ~nooijen/chem440/molpro/n_atom. The following is the content of the file cc_ground_4_s.inp ***,N atom ground state: quartet S basis=cc-pvdz! title {rhf; uccsd(t); The WF keyword specifies the numer of electrons (7), the spatial D 2h irrep of the state (block 8) and the spin-multiplicity (2S=3). From the output we can get the relevant results: UCCSD(T) RHF-SCF We can run the same calculation using the MRCI approach. Here is the input file: ***,n basis=cc-pvdz {rhf;! title 5

6 6 {casscf;! do casscf calculaion! define wavefunction symmetry! do mrci calculaion! define wavefunction symmetry To run the MRCI calculation we first need to run a CASSCF calculation in general. MRCI includes a more complete description of electron correlation and is essential to get correct results. Here is the final output: MRCI CASSCF RHF-SCF You see that in this case the CASSCF and RHF=ROHF results coincide, because of the half-open shell nature of the state. The MRCI result is far more accurate as it includes electron correlation effects. You can also find the following section in the output Number of closed-shell orbitals: 1 ( ) Number of active orbitals: 4 ( ) Number of external orbitals: 9 ( ) The closed-shell orbitals refer to the 1s orbitals for the nitrogen atom. The active orbitals refer to the valence orbitals, 2s and 2p, and it lists them by number for each symmetry block. The external orbitals refer to the virtual orbitals, and this depends on the size of the basis set, which is only cc-pvdz here. Just below this output you find State symmetry 1 Number of electrons: 5 Spin symmetry=quartet Space symmetry=8 Number of states: 1 Number of CSFs: 1 (1 determinants, 4 intermediate states) Here the number of electrons is the # of active electrons (2s and 2p). It indicates we are interested in the quartet state of symmetry block 8. We have full control over all of this in the input. Let me indicate how this is done, using the following input file mrci_4_s_def.inp in which we define this information in the input file itself. 6

7 7 ***,n basis=cc-pvdz {rhf;! title! rohf calculation {casscf;! do casscf calculation OCC,2,1,1,0,1,0,0,0;! specify total occupied space FROZEN,1,0,0,0,0,0,0,0;! specify Frozen core orbitals! define wavefunction symmetry ecasscf=energy! do mrci calculation OCC,2,1,1,0,1,0,0,0;! specify occupied space CORE,1,0,0,0,0,0,0,0;! specify frozen core orbitals! define wavefunction symmetry e_4_s=energy(1)! MRCI energy eq_4_s=energd0(1)! MRCI+Q energy (Davidson correction). {aqcc;! do mraqcc calculation OCC,2,1,1,0,1,0,0,0;! specify occupied space CORE,1,0,0,0,0,0,0,0;! specify frozen core orbitals! define wavefunction symmetry {acpf;! do mracpf calculaion OCC,2,1,1,0,1,0,0,0;! specify occupied space CORE,1,0,0,0,0,0,0,0;! specify frozen core orbitals! define wavefunction symmetry We did quite a bit more here. In the CASSCF section I defined the 1s orbitals to be frozen. This means they are not optimized, but taken from the ROHF calculation. In the MRCI section I specified CORE= This means that these orbitals are not included in the treatment for electron correlation (like DROPMO= in ACES2). In the MRCI section I picked up both the MRCI and the MRCI+Q energies, as indicated. I did two more calculations, which are very similar to MRCI+Q. They are attempts to get more accurate energies and are referred to as the MR-ACPF and MR-AQCC approaches in the literature. I am just listing them here, as they are quite useful in practice. This input can be made a bit more compact for future reference. Here is the input file method_4_s.inp 7

8 8 ***,n! title basis=cc-pvdz methods=[mrci,aqcc,acpf] {rhf; {casscf;! do casscf calculaion OCC,2,1,1,0,1,0,0,0;! specify occupied space FROZEN,1,0,0,0,0,0,0,0;! specify Frozen core orbitals! define wavefunction symmetry! Calculate the ground state: quartet S do i=1,#methods method=$methods(i) {$method; OCC,2,1,1,0,1,0,0,0; CLOSED,1,0,0,0,0,0,0,0; CORE,1,0,0,0,0,0,0,0; E(i)=energy(1) E_Q(i)=energd0(1)! specify occupied space! specify closed-shell (inactive) orbitals! specify frozen core orbitals enddo! tabulate results for various methods {table, methods, E, E_Q head, method, E, E+Q save,method_4_s.tab title, total energies for N atom Quartet S state I specified a number of methods, using methods=[mrci,aqcc,acpf], created a loop over the methods and then made a table for each. This shows some of the flexibility of the Molpro input. I am impressed! Let us now attempt to get all of the valence type excited states for the Nitrogen atom. I used a common set of orbitals, corresponding to the half-filled 4 S state, which has equal population of px, py, p z and hence preserves the spherical symmetry of the orbitals. The input file N_atom/mrci_all.inp provides a (lengthy) input file to do this: 8

9 9 ***,N! title basis=cc-pvdz methods=[mrci] sym=[1,2,3,4,5,6,7,8,9] {rhf; {casscf;! do casscf calculaion OCC,2,1,1,0,1,0,0,0;! specify occupied space FROZEN,1,0,0,0,0,0,0,0;! specify Frozen core orbitals! define wavefunction symmetry! Calculate the ground state: quartet S do i=1,#methods method=$methods(i) {$method; OCC,2,1,1,0,1,0,0,0; CLOSED,1,0,0,0,0,0,0,0; CORE,1,0,0,0,0,0,0,0; E(8)=energy(1) E_Q(8)=energd0(1)! specify occupied space! specify closed-shell (inactive) orbitals! specify frozen core orbitals! Doublet D states (5 in total) {$method WF,7,4,1;! define wavefunction symmetry E(4)=energy(1) E_Q(4)=energd0(1) WF,7,6,1; E(6)=energy(1) E_Q(6)=energd0(1) WF,7,7,1; E(7)=energy(1) E_Q(7)=energd0(1) WF,7,1,1; state,2;! specifies # of states in this symmetry block 9

10 10 E(1)=energy(1) E_Q(1)=energd0(1) E(9)=energy(2) E_Q(9)=energd0(2)!# Doublet P states WF,7,2,1; E(2)=energy(1) E_Q(2)=energd0(1) WF,7,3,1; E(3)=energy(1) E_Q(3)=energd0(1) WF,7,5,1; E(5)=energy(1) E_Q(5)=energd0(1) enddo {table, sym, E, E_Q head, symmetry, MRCI, MRCI+Q save,mrci_all.tab title, MRCI total energies for N atom ground and excited states sort,2,1,3 As you can see, I am asking for many MRCI calculations, all based on the same CASSCF orbitals optimized for the 4 S state. This yields the following table (mrci_all.tab) MRCI total energies for N atom ground and excited states SYMMETRY MRCI MRCI+Q You can see one state with the lowest energy in symmetry block 8. This is the 4 S state as before. Then three degenerate states which comprise the 2 P multiplet. Finally 5 states, which are almost degenerate make up the 2 D multiplet. In reality they are exactly 10

11 11 degenerate, and this slight shift (0.1 mh) is a feature of Molpro. The above calculation is not sufficiently accurate, because we used the orbitals of the 4 S state for all of the other states too. I did this, because I wanted to determine the degeneracy of each state. Now we know. You can also verify that this information on the symmetry agrees with the prior discussion based on the irreducible representation table on page 4. In block 8, 2S=3 we find the 4 S state, in blocks 2, 3 and 5, 2S=1, we find the 2 P states, while in blocks 4, 6, and 7, 2S=1 we have easy access to the 2 D states. Two more states are found in block 1, 2S=1. Now I can target these states specifically. Let us run the calculations specified in methods_2_d.inp: ***,n! title basis=cc-pvdz $methods=[mrci,aqcc,acpf] {rhf;! use the half-open shell state {casscf;! do casscf calculation OCC,2,1,1,0,1,0,0,0;! specify occupied space FROZEN,1,0,0,0,0,0,0,0;! specify Frozen core orbitals WF,7,4,1;! define wavefunction symmetry! Calculate the doublet D state in symmetry block 4 do i=1,#methods $method=$methods(i) {$method; OCC,2,1,1,0,1,0,0,0; CLOSED,1,0,0,0,0,0,0,0; CORE,1,0,0,0,0,0,0,0; WF,7,4,1; E(i)=energy(1) E_Q(i)=energd0(1)! specify occupied space! specify closed-shell (inactive) orbitals! specify frozen core orbitals Enddo! tabulate results for various methods {table, methods, E, E_Q head, method, E, E+Q save,method_2_d.tab title, total energies for N atom Doublet D state 11

12 12 I optimize the orbitals specifically for this state: WFN, 7, 4, 1, and then run MRCI, MRCI+Q, MRACPF and MRAQCC calculations (using the loop over methods). Here are the results collected in Table method_2_d.tab total energies for N atom Doublet D state METHOD E E+Q MRCI AQCC ACPF We can run a similar calculation for the 2 P (method_2_p.inp). The calculations thus far use a fairly small basis set. You can increase the basis set, for example to cc-pvqz. Then the excitation energies can be compared to experimental results. The experiments include effects due to spin-orbit coupling, which are not included in the present calculations. I took the results from the NIST web site on atomic energy levels (see also ACES2 labs). Low lying atomic energy levels for Nitrogen atom (in cm-1). 2s 2 2p 3 4 S 3 / 2 0 2s 2 2p 3 2 D 5 / / s 2 2p 3 2 P 1 / / You can compare the results of your calculations to experiment (convert energy differences from Hartree to cm -1 ). 3. Specifying geometries for molecules in molpro. There are a number of ways to specify geometries in Molpro. A method that is particularly convenient is to import a geometry file with Cartesian coordinates obtained from another program (e.g. Gaussian or a database). Below is provided the input for a calculation on the ground state of benzene. In the Molpro/benzene directory you can find the file benzene.inp. You should see this: 12

13 13 ***,benzene! title geometry=benzene.xyz rhf casscf mrci This file may seem like it should have more information. It appears that the orbital occupancy and the closed shell orbitals have not been specified. The Hartree-Fock calculation generates these directives for the CASSCF and MRCI calculation. The geometry is specified as another file located in this directory. In the input file you are telling Molpro it is in the file benzene.xyz, which contains the information below: 12 benzene.xyz C C C C C C H H H H H H The number at the top of the document specifies the number of atoms. Below this you will find the file name and four columns. The first column tells Molpro what atom is being specified and the next three columns specify the x, y, and z coordinates in Ångstrom. This is one way to specify geometries. Read the online Molpro manual if you wish to know more about this beautiful program! 13