Practical Issues on the Use of the CASPT2/CASSCF Method in Modeling Photochemistry: the Selection and Protection of an Active Space Roland Lindh Dept. of Chemistry Ångström The Theoretical Chemistry Programme Uppsala University Sweden The Winter School, Helsinki, Suomi 12-16 December 2011
General Outline Lecture 1 CASSCF/CASPT2 basics Lecture 2 two specific molecular studies
Outline A Tutorial The CASSCF/CASPT2 Paradigm General rules for selecting an active space Orbital representations: atomic orbitals Orbital representations: localized molecular orbitals Same-but-different: purpose designed active space Dirty tricks: Active space stabilization
The CASPT2/CASSCF Paradigm Is a single determinant wave function enough? No! Because it can not: - in general describe a Bond Formation/Breaking - describe a Transition State Structure - describe a Conical Intersection - describe a singlet biradical structure - on equal footing describe several states at the same time. (Photo)Chemistry include many situations in which two or several electronic configurations are near or exactly degenerate. We need a method which can simulate this.
Breaking the H2 bond u r =N u 1s A r 1s B r g r = N g 1s A r 1s B r
H2 wave functions The molecular orbitals: σ g ( r )=N g (1s A (r )+1s B (r )) σ u (r )=N u (1s A (r ) 1s B (r )) The SCF wave function: = g r 1 g r 2 2,0 =N 2g 1s A r 1 1s A r 2 1s A r 1 1s B r 2 1s B r 1 1s A r 2 1s B r 1 1s B r 2 2,0 The single configuration wave function contains + + both terms as H + H, H + H and H + H in a fixed ratio! We need some flexibility. The CAS wave function: =( C 1 1s A r 1 1s A r 2 C2 1s A r 1 1s B r 2 C 3 1s B r 1 1s A r 2 C 4 1s B r 1 1s B r 2 ) 2,0
H2 the CASSCF way: 2-in-2 The CASSCF wave function has the correct asymptotic behavior!
The Complete Active Space SCF The CASSCF model : Inactive orbitals Active space orbitals Virtual orbitals The CASSCF is a Full-CI in a subspace of the orbital space. It is a spin eigen function. The (SA-)CASSCF model treats the static correlation. For qualitative accuracy add ee correlation with perturbation (MS)-CASPT2. Natural orbital analysis gives partial occupation numbers (0-2). State Average CASSCF treats several states at the same time.
The Restricted Active Space SCF
The active space Select active orbitals to: Give correct dissociation Correct degeneracies ( incomplete shells) Correct excited states Treat near-degeneracies The orbitals around the Fermi gap are the candidates (be careful!!!). Perfect pairs: σ-σ*, π-π*, δ-δ* Lone pairs (n): maybe.
General rules for selecting active orbitals: atoms and atomic ions nd 2 row elements: 2s and 2p (more than 4 valence electrons skip the 2s).
General rules for selecting active orbitals: atoms and atomic ions rd 3 row elements: 3s and 3p (more than 3 valence electrons skip the 3s). As the spx hybridization is reduced down the periodic table do not include the s-shell.
General rules for selecting active orbitals: atoms and atomic ions st 1 Transition Metals: 4s, 3d and 4p (more than 5 d-electrons might need 4d double-shell effect). For higher row TMs the double-shell effect is reduced.
General rules for selecting active orbitals: atoms and atomic ions Lanthanides: 4f, 6s, 6p and 5d Actinides: 5f, 7s, 7p and 6d Be careful wrt double-shell effects for the f-orbitals
General rules for selecting active orbitals: atoms and atomic ions For Rydberg states: include these in the active space. Note: use Rydberg specific basis sets!
General rules for selecting active orbitals: Molecules Look for: correlating pairs : σ-σ*, π-π*, etc. orbitals of the excited state: n and Rydberg equivalent partners What is the process we are studying? Some sloppy rules: CH bonds can be inactive All p orbitals in unsaturated molecule Rydberg orbitals for excited states above 5 ev
Orbital Representation After we have selected the active space (x-in-y), which is an intellectual challenge, we have to generate it, this is more of a technical challenge!
Orbital Representations: atomic orbitals
Orbital Representations: localized molecular orbitals Localized occupied and virtual SCF orbitals, respectively.
Orbital Representations: the SCF orbitals The SCF orbitals are delocalized! Virtual orbitals are not well defined! Note that these orbitals are from a minimal basis calculation.
Orbital Representations: the SCF orbitals The SCF orbitals are delocalized! Useless for localized processes (e.g. H abstraction)
Orbital Representations: the virtual SCF orbitals Virtual Orbitals are not well define! The six lowest virtual SCF π orbitals in a triple-ς basis.
Orbital representations: Strategy Use localized orbitals, AOs or MOs Do never use SCF orbitals Start with a MB basis and expand Double check all the time! Protect your orbitals once you have found them Be paranoid
Same-but-Different The MCSCF solution to a specific active space is not unique! Demonstration: For butadiene we would like to study the fragmentation process of: a) C4H6 2 C2H3 b) C4H6 C3H4 + CH2 In both cases we will have the 4 π orbitals active together with the correlating pair of the bond which we are breaking (σ-σ*), that is a 6-in-6 CAS in both cases.
Same-but-Different: The Starting Orbitals The π-space: The correlating pairs in the σ-space: C4H6 2 C2H3 C4H6 C3H4 + CH2 By carefully selecting the starting orbitals I select the most likely CASSCF solution.
Same-but-Different: The CASSCF Orbitals C4H6 2 C2H3 Bingo! C4H6 C3H4 + CH2 Note the active space on the right does not preserve equivalent methyl bonds.
Dirty Tricks: Active Space Stabilization The mathematical (local) solution to the MCSCF equations are those that have strong correlating pairs. The σ bonds and lone pairs are the general problems.
Dirty Tricks: Two near-degeneracies destabilize the (local) mathematical solution: Active orbitals with an occupation close to 2 Active orbitals with an occupation close to 0 The orbitals can slip into the inactive or the virtual space, respectively. We need to have a mathematical model for which the occupation number are not close to 2 or 0!
Dirty Tricks: The solution to the problem is SA-CASSCF! The SA-CASSCF occupation numbers depend on the average occupation numbers of all the states that are included in the calculation. -Stabilize σ orbitals by including a state in which you have some excitation out of the σ orbital. -Stabilize n orbitals by including a state in which you have some (n-σ* or) n-π* excitation. This trick is only possible if you start with the correct active orbital manifold!!!
Summary part I Why Multi-configurational Methods The CASSCF model The active space Different orbital representations Be careful with SCF orbitals - expand Standard AO active orbitals MO active orbitals Localized starting MO orbitals The solution to the MSCF eq. is not unique!!! Tricks Dynamical Electron correlation and Dispersion with (MS-)CASPT2
Part II: Explicit Cases, the organic process Case 1: water Case 2: para-nitroaniline
Why Water? Possible valence excited states are only n-σ* and σ-σ* excitations. Rydberg states are important early on. Design of Rydberg basis set. An example in which the problem with state reordering between CASSCF and CASPT2 occurs.
What are Rydberg orbitals? The orbitals of the excited states of atomic hydrogen is the origin of the concept of the so-called Rydberg orbitals. Victor Rydberg
What are Rydberg orbitals? The series of bound states in hydrogen describe a progression towards a description of the electron as being completely removed. The binding energy, Eb, is computed as E b= Ry n 2 ; R y =13.61 ev That is, the series of Rydberg states represents a progression of steps toward the ultimate ionization of the hydrogen atom.
What are Rydberg orbitals? The ionization of a molecular system will at the limit of be identical to the ionization process of hydrogen. At the limit of ionization the molecular Rydberg orbitals will be similar to the hydrogen Rydberg orbitals. The molecule behaves as a superatom. In a molecular system the Rydberg orbitals are of molecular origin, not atomic. The Rydberg orbitals are diffuse and not localized to a particular atom.
Designing the Rydberg basis. Universal Gaussian-Basis Sets for an Optimum Representation of Rydberg and Continuum Wavefunctions K. Kaufmann, W, Baumeiter and M. Jungen, J. of Phys. B, 22:223-2240 (1989). The procedure of Kaufmann et al. generates a series of Gaussian exponents which are used in subsequent design of the molecular Rydberg basis set. More about this later. Current standards for molecular systems use 8s8p8d exponents.
Some comments about origin of optically excited states in neutral molecules The excited states are generated by solutions to Schrödinger's equation which has different occupation numbers that those of the ground state. These can be classified as single, double, etc. excited states. The singly excites states can be pictured as moving a single electron from an occupied orbital to a virtual orbitals. In the case of degeneracies a singlet excited state is depicted as a linear combination of such single excitations.
Some comments about origin of optically excited states in neutral molecules The virtual orbitals are of three types: Valence orbitals (σ*, π*, δ*,..) Rydberg orbitals Orbital corresponding to atomic excited orbitals (for example, in carbon 3s, 3p, etc...) These differences should guide us in the way in which we describe these orbitals when we do calculations of excited states.
Some comments about origin of optically excited states in neutral molecules The lowest excited states correspond to a single electron change of the occupation of the orbitals around the Fermi gap (n-π*, π-π*,...) We note that some cases of possible valence excited states will be in the continuum (could be meta-stable), for example states for which an electron is moved to an σ* orbital usually corresponds to a state above the IP. For the very same reason excited states corresponding to exciting electrons to excited atomic orbitals have normally energies above the IP (possibly observed as Fano resonances).
Suggestions To improve the description of the valence excited states increase the flexibility of valence basis set to modulate the valence orbitals. Rydberg orbitals are molecular orbitals and should not be generated via adding diffuse functions to the atomic functions (leads fast to large basis set and linear dependence). Forget about atomic excited orbitals beyond the valence orbitals in molecular systems.
Designing the Rydberg basis for water. The Rydberg basis is a pseudo atomic basis set! 1. Determine the position of the pseudo atomic center. 2. Compute the Rydberg orbitals 3. Generate the Rydberg molecular basis set.
Designing the Rydberg basis for water Compute the center of charge for H 2O+ with either UHF or CASSCF.
Designing the Rydberg basis for water. Add the uncontracted Kaufmann basis to the water cation calculation and recompute. The unoccupied orbitals with negative orbital energies represents the origin of the Rydberg basis set.
Designing the Rydberg basis for water. Isosurface @ 0.040 au Isosurface @ 0.015 au 3s 3p
Designing the Rydberg basis for water. Isosurface @ 0.015 au 3d
Designing the Rydberg basis for water. Note that the final results are not critically dependent on the center of the Rydberg basis. For example: A sp Rydberg basis set will allow for polarization of the 3s Rydberg orbital A spd Rydberg basis set will allow for polarization of the 3s and 3p Rydberg orbitals This flexibility render the finer details of the center of the Rydberg basis to be insignificant.
Water optical excited states: the active space The active space is selected as a combined full-valence and Rydberg (3s & 3p) space. σ and σ* lone-pair
Water Optical excited states: vertical excitation energies (ev) State ΔE f 11A1 ΔE f ΔE f Expt - 11B1, n 3s/σ* 7.39, (7.39) 0.046 7.4, 7.447, 7.42 11A2, n 3py/σ * 9.14, (9.14) forbidden 9.1 21A1, n 3s/σ* 9.63, (9.63) 0.063 9.7, 9.991 21B1, n 3pz 9.78, (9.78) 0.008 10.01, 9.99 31A1, n 3px 9.96, (9.96) 0.086 10.16, 10.17
The Reduced Active Space? Considering that the bonding s orbitals are not principal orbitals of the lowest excited states can we move them outside the active space and get the same result. How much can the CASPT2 recover the difference?
Water Optical excited states: vertical excitation energies (ev) State ΔE f ΔE f 11A1 ΔE f Expt - 11B1, n 3s/σ* 7.39 0.046 11A2, n 3py/σ * 9.14 21A1, n 3s/σ* 7.48 0.049 7.4, 7.447, 7.42 forbidden 9.24 forbidden 9.1 9.63 0.063 9.74 0.082 9.7, 9.991 21B1, n 3pz 9.78 0.008 9.89 0.007 10.01, 9.99 31A1, n 3px 9.96 0.086 19.02 0.067 10.16, 10.17
The Restricted Active Space SCF To what extent do we need a CAS to model the Rydberg states? Could we move these to the RAS3 space/ 1particle? The subsequent RASPT2 will try to recover the missing contributions as compared to a CASSCF reference.
The parameter size full-valence CI, Rydberg : 13860 CSFs full-valence CI σ,rydberg: 336 CSFs full-valence CI, Rydberg in RAS3: 945 CSFs full-valence CI σ, Rydberg in RAS3: 100 CSFs
Water Optical excited states: vertical excitation energies (ev) State ΔE f ΔE f ΔE 11A1 f Expt - 11B1, 7.39 0.046 7.48 0.049 7.40, (7.49) 11A2, 9.14-9.24-9.17, (9.27) 21A1, 9.63 0.063 9.74 0.082 9.77, (9.74) 0.053 9.7, 9.991 21B1, 9.78 0.008 9.89 0.007 9.83, (9.88) 0.008 10.01, 9.99 31A1, 9.96 0.086 19.02 0.067 9.98, (10.04) 0.078 10.16, 10.17 n 3s/σ* n 3py/σ* n 3s/σ* n 3pz n 3px 0.040 7.4, 7.447, 7.42 9.1
Para-nitroaniline Why para-nitroaniline? A part from the scientific interest of this system (solvatochromic shift of the CT state) the computational challenges are A bit larger (as compared to water) P-p* manifold Mixing of Rydberg and Valence states Changed order of states CASSCF vs CASPT2 Amine group: electron donar Nitro group: electron acceptor
Para-nitroaniline: the active space Amine group: p-orbital (1o,1e) Benzene group: p system (6o,6e) Nitro group: p-orbitals(3o,5e) and lone-pairs (2o,4e) Rydberg orbitals (3s and 3p) (4o,0e) Active space (12+4)o 16e Valence only CI (12o,16e); 70785 CSFs
Active orbitals: the valence space The nitro lone-pairs not stable in the active space for single root CASSCF. How many roots do we need minimum?
Active orbitals: the valence space / 3 root SA-CASSCF The two lowest Excited states are: n-π*
Rydberg basis: basis set position The Rydberg basis is contracted 8s8p8d to 2s2p2d.
Para-aniline This system could be done with CASSCF/CASPT2, (16e,16o). However, we could get around with a RASSCF/RASPT2 model (429,660 CSFs) as follows RAS1: the oxygen lone-pairs RAS2: the π-system RAS3: the Rydberg orbitals (single particle approx.)
Order of States In the CASSCF the π π* states, as compared to the n R states, are disfavored. Make sure that the CASSCF roots include what you are looking for! List below depict the state reordering in pna. 1 2 3 4 5 6 7 8 1 2 3 4-6+9 5 8 7 9 10 11 12 13 14 15 16 6+9 13 12 10 14 15 16 11
Results
Summary Part II Parts of the organic process of doing CASSCF/CASPT2 calculations Active space Valence vs. Rydberg states Rydberg basis sets RASSCF/RASPT2 approximation Accuracy 0.1-0.2 ev is expected Benchmarking and understanding