The MCSCF Method *, Molecular Orbitals, Reference Spaces and COLUMBUS Input

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1 The MCSCF Method *, Molecular Orbitals, Reference Spaces and COLUMBUS Input Hans Lischka University of Vienna *Excerpt of a course presented by R. Shepard, Argonne National Laboratory, at the Workshop on Theoretical Chemistry, Feb. 2004, Mariapfarr, Austria 1

2 MCSCF Approach Single-configuration wave function: Multi-configuration wave function: Ψ Ψ SR MR = = Φ Φ 1 1 K c I Ψ Φ Φ I i i { Φ } i KΦ n Φ n In a MCSCF calculation two types of optimizations have to be performed: (i) (ii) w.r.t. the expansion coefficients of the CSFs and w.r.t. the MO coefficients 2

3 Orbital Variation Exponential parametrization of the transformation matrix Φ = Φ 0 U=χC 0 U with fixed C 0 such that C 0T SC 0 =1 U must be orthogonal Let U=exp(K) with K= K T => N(N-1)/2 independent unconstrained parameters K pq (e.g. for p>q) Kˆ = = r, s r> s K K rs rs E Tˆ rs rs = r> s K mc; Φ = exp( Kˆ ) mc; Φ rs ( E E ) 0 rs sr 3

4 CSF Variation Ncsf 1 P ˆ = p ( n n 0 0 n ) = p ˆ n n=1 Ncsf Ncsf 1 n=1 0 = c n0 n, m = c nm n with m 0 = 0 n=1 mc = exp( ˆ P )0 Ncsf n=1 P n ρ = mc ˆ H mc = 0 exp( ˆ P ) ˆ H exp( ˆ P )0 4

5 Combined Orbital and CSF Variation mc;φ = exp( ˆ K ) mc;φ 0 = exp( ˆ K )exp( ˆ P )0;φ 0 ρ = mc;φ ˆ H mc;φ = mc;φ 0 exp( ˆ K ) ˆ H exp( ˆ K ) mc;φ 0 = 0;φ 0 exp( ˆ P )exp( ˆ K ) ˆ H exp( ˆ K )exp( ˆ P )0;φ 0 5

6 Energy Variation First derivatives ρ K rs K= 0 p= 0 ρ p n K= 0 p= 0 w rs = 0;φ 0 v n = 0;φ 0 [ H ˆ, T ˆ rs ]0;φ 0 [ H ˆ, P ˆ n ]0;φ 0 6

7 Energy Variation II Second derivatives 2 ρ K pq K rs K= 0 2 ρ p= 0 p m p n K= 0 p= 0 [ T ] + ˆ rs B pq,rs = 1 0;φ 0 [ H ˆ, T ˆ 2 ], ˆ pq [ P ] + ˆ n M mn = 1 0;φ 0 [ H ˆ, P ˆ 2 ], ˆ m [ H, T ˆ ], T ˆ ] 0;φ 0 rs pq [ H, P ˆ ], P ˆ ] 0;φ 0 n m 2 ρ K pq p n K= 0 p= 0 C pq,n = 0;φ 0 [ H ˆ, T ˆ ], P ˆ ] 0;φ 0 pq n 7

8 8 Energy Derivatives III ( ) ( ) ( ) = = p k M C C B p k v w p k Cp k Mp p Bk k v p w k p k T T T T T T T T T T E E,,, ρ Newton-Raphson type procedure (second order convergence) Improvements of convergence radius (estimate of higher order terms)

9 Molecular Orbitals and MCSCF The correct form of the MOs is crucial for the subsequent CI calculation Balancing several electronic states simultaneously state averaging (SA) Choice of the orbital and CSF spaces Complete Active Space (CAS) all possible CSFs are constructed within a given space Restricted Active Space (RAS) only n-tuple holes are allowed (n-fold excitations from doubly occupied space) Auxiliary space (AUX) n-fold excitations into an origianlly empty orbital space 9

10 Orbital Spaces AUX CAS n-electrons RAS n-tuple excitations 10

11 Orbital Spaces II Restricted direct product spaces Generalized Valence Bond Perfect Pairing (GVB-PP) wavefunction 2 el. in 2 orb. n Each electron pair is restricted to singlet coupling localized orbitals 11

12 Pyrrole 12

13 Pyrrole II 13

14 $COLUMBUS/colinp: Columbus Input COLUMBUS INPUT FACILITY main menu options -> 1) Integral program input (for argos/dalton/turbocol) 2) SCF input 3) MCSCF input 4) CI input 5) Set up job control 6) Exit the input facility 14

15 colinp input I Integral input: Cartesian geometry, choice of basis set from library, select Dalton integral program SCF input: number of occupied orbital per irred. rep. MCSCF input: Enter number of DRTS [1-8]: The distinct row table describes the wave function in the unitary group approach (UGA). For each class of wavefunctions (e.g. for each symmetry, for each multiplicity) we need one DRT. number of electrons for DRT #1 multiplicity for DRT #1 spatial symmetry for DRT #1: irreps are numbered (according to ordering in AO integral section) 15

16 colinp input II excitation level (cas,ras)->aux: 0 excitation level ras->(cas,aux): 0 number of doubly occupied orbitals per irrep:?? number of CAS orbitals per irrep:?? Apply add. group restrictions for DRT 1 [y n]: n Input for characterization of wavefunction finished Input for MCSCF procedure: state-averaging! 16

17 Ethylene CAS(2,2) Two SCF orbitals (one closed shell, occupied and one virtual orbital) are moved to the CAS CAS(2el,2orb) Closed shell SCF CAS(2el,2orb): 1b 2g 1b 3u π* orbital: virtual π orbital σ orbitals: (14 el., 7 orb.) Doubly occ. (DOCC) 1b 2g 1b 3u π-cas DOCC (14el,7orb) 17

18 Ethylene Orbitals 1a g 1b 1u 2a g 1b 2u 2b 1u y 3a g z 18

19 Ethylene Orbitals II 1b 3g 1b 3u (π) 1b 2g (π*) 19

20 Representation as symmetry table for input into COLUMBUS SM1 Orbital occupation of ethylene CAS(2,2), D2h System: C2H4, Point Group: D2h, N. Electrons: 16, Multiplicity: 1 Level: MR-CISD/SA-3-CAS(2,2) IRREP a g b 3u b 2u b 1g b 1u b 2g b 3g a u SCF DOCC 3 1 (π) MCSCF DOCC RAS CAS 0 1 (π) (π*) 0 0 AUX MRCI FC FV DOCC ACT AUX INT

21 State Multiplicity N. electrons Symmetry (A g ) (π) (B 1u ) (π) 1 (π ) (A g ) (π ) 2 Number of distinct row tables (DRTs): 2 21

22 colinp input III 1) Def. of CI wave function (data from MCSCF) 2) Def. of CI wave function for geom.opt (data from MCSCF) 3) Def. of CI wave function - one-drt case 4) Def. of CI wave function - one-drt case (geom.opt.) 5) Definition of CI wave function - multiple-drt case 6) Definition of CI wave function - multiple-drt case (NAD coupl.) 7) Skip DRT input (old input files in the current directory) 22

23 colinp input IV Input for the reference wavefunction: procedure similar to MCSCF case number of frozen core orbitals per irrep number of frozen virt. orbitals per irrep: 0 0 number of internal(=docc+active+aux) orbitals per irrep ref doubly occ orbitals per irrep auxiliary internal orbitals per irrep Enter the excitation level (0,1,2): 2 Generalized interacting space restrictions [y n]: y Enter the allowed reference symmetries: accept default Apply additional group restrictions for DRT [y n]: n 23

24 Ethylene: MR-CISD input: CAS(2,2) reference space This reference space is identical to the CAS(2,2) of the MCSCF calculation. Core orbitals (1a g and 1b 1u are moved to the frozen core (FC) space All single and double excitations are constructed from all references IRREP a g b 3u b 2u b 1g b 1u b 2g b 3g a u SCF DOCC 3 1 (π) MCSCF DOCC RAS CAS 0 1 (π) (π*) 0 0 AUX MRCI FC FV DOCC ACT 0 1 (π) (π*) 0 0 AUX INT

25 colinp input V Type of calculation: CI [Y] AQCC [N] AQCC-LRT [N] LRT shift: LRTSHIFT [0 ] State(s) to be optimized NROOT [1 ] ROOT TO FOLLOW [0] Reference space diagonalization INCORE[Y] NITER [ ] RTOL [ ] Bk-procedure: NITER [1 ] MINSUB [1 ] MAXSUB [6 RTOL [1e-3 ] CI/AQCC procedure: NITER [20 ] MINSUB [1 ] MAXSUB [6 ] RTOL [1e-3 ] FINISHED [ ] 25

26 colinp input VI 5) Set up job control: 1) Job control for single point or gradient calculation 2) Potential energy curve for one int. coordinate 3) Vibrational frequencies and force constants 4) Exit 1) single point calculation 26

27 colinp input VII 1) (Done with selections) 2) [ ] SCF 3) [ ] MCSCF 4) [ ] transition moments for MCSCF 5) [ ] standard MR-CISD with one DRT (ciudg) 6) [ ] standard MR-CISD with several DRTs (ciudg) 7) [ ] transition moments for MR-CISD 8) [ ] <L> value calculation for MR-CISD 9) [ ] parallel MR-CISD (pciudg) 10) [ ] sequential version of pciudg (sciudg) 11) [ ] one-electron properties for all methods 12) [ ] convert MOs into molden format 13) [ ] finite field calculation for all methods 27

28 Executing Columbus & Results $COLUMBUS/runc m >&runls & This command will execute a Columbus calculation according to the specifications given in the colinp input. Note: Columbus consists of a collection of individual programs communicating via files. Getting results: see directories LISTINGS, MOLDEN and MOCOEFS (see WORK for actual calculation) Each program step usually has its own listing: e.g. mcscfls, mcscfsm; ciudgls, ciudgsm, More information under: ntation_main.html item usage 28