Internally Contracted Multireference Coupled Cluster Theory. Computer Aided Implementation of Advanced Electronic Structure Methods.

Transcription

1 Internally Contracted Multireference Coupled Cluster Theory. Computer Aided Implementation of Advanced Electronic Structure Methods. Marcel Nooijen Chemistry Department University of Waterloo

2 Summary In the past decade efficient Coupled Cluster methods have been developed to obtain accurate results for electronically excited states, e.g. Equation-of-Motion CC (EOMCC) / CC Linear Response Theory (CCLRT) / SAC-CI, Fock Space CC and Similarity Transformed EOMCC (STEOM-CC). All of these methods can be characterized as "Transform & Diagonalize, T&D" approaches: Single reference techniques are used to extract a transformation of the Hamiltonian that incorporates dynamical correlation effects. The transformed Hamiltonian is subsequently diagonalized over a compact subspace to obtain the states of interest. The focus of the talk will be on a generalization of the T&D concepts to genuine multireference situations. This leads to an efficient formulation of a state selective internally contracted multireference CC theory and I will present some applications to medium sized systems. Although the methodology is efficient, it is also rather complicated. We developed an automation tool, the Automatic Program Generator (APG), that allows us to derive, factorize and implement equations in an effective way. The APG has been invaluable to develop our current multireference coupled cluster strategy, as it allowed us to explore quickly a large number of variations on the theme.

3 Basic Strategy: "Transform then Diagonalize". Transform: G = U 1 HU - Transformation U should account for short-range, dynamical, electron correlation effects. - Can be expected to work for multiple states at once, if they are in some sense similar. - Transformation does not change eigenvalues. Diagonalize: GC = CE - Choose U such that C can be compact. Truncated diagonalization space / selection of configurations. Advantages: - Access to wave function. - Multiple states may be treated at once balance. - Approach is in principle exact, independent of U. Disadvantages: - Selection of diagonalization space (active space) will depend on user. Unlikely to be completely black box. - Transformation U is in principle arbitrary. How to find something that is useful in practice?

4 Internally Contracted MRCC Theory Basic form for the wave function T Ψ = e ( 1+ S ) R 0 0 : Closed shell determinant (vacuum in MB theory). R 0 : Reference state qualitatively correct wfn. T = T + T 1 2 excitation operators w.r.t. 0 (q-creation). S = S + S 1 2 : S 0 = 0, but SR 0 0. (q-annihilation). - Spin- and -symmetry adaptation: T and S have same symmetry as for closed shell case (generators of unitary group). - Size-consistency: e T suffices (because S 0 = 0) - Internally Contracted:, T S act as an entity on reference state relatively few parameters. - State selective: Coefficients will be determined for one state at a time (or a few states in a state averaged formulation).

5 Define ( 1+ S ) R = C Ψ = e T C 0 Schrödinger equation : T He C T 0 = Ee C 0 T T e He C 0 = HC 0 = EC 0 Akin to Equation of Motion CC Theory. In EOM-CC e T 0 itself is approximate eigenstate Here T (and 0 ) are specific for state of interest. Define left-hand zeroth-order eigenstate 0 L 0 ab T T La e He C 0 0La ab C ij = ij 0E = 0 T 2 Φ T T e He C 0 = E Φ C 0 = Ec C λ λ λ Equation for single excitations?? What is 0?? Is C particle conserving??

6 Scheme I : Core-vacuum (p-type CI) abcd,,, n, virtual v pqrs,,, np, partial ( active) i, j, k, l n, occupied o R S T AB R S T A U V W ABC,,, RAB AB S AB T AB N A Φ core A AB T = t a a + t a aa a ~ N n n n + 2n n n i A i ij A i B j A o v o v p o C= caba Aa B + c a a a a ~ ABCk ABCk A B C k ( NA) AB,,,, 3 n o Relatively few coefficients in exponential. Very many in operator C

7 Scheme II: Full Vacuum (h-type CI) abcd,,, n, virtual v pqrs,,, np, partial ( active) i, j, k, l n, occupied o R S T R S T R S T A B A AB AB AB AB U V W Φ IJK,,, a ab T = ti a a a I + tij a a a I a b a J ~ nn nn + 2nnn 3 C= c a a + c a a a a ~ Nn IJ I J IJcK I J c K full N I v I v o o p v I v Many coefficients in exponential operator T. Relatively few in operator C

8 Scheme III: half filled vacuum (# of electrons). ph-type CI - abcd,,, n, virtual v pqrs,,, n, partial p ABC,,, A U ( active) i, j, k, l n, occupied o R S T R B S T A m RAB AB S AB T AB U V W V W IJK,,, N N I A Φ 0 A AB T = t a a + t a a a a ~ n N n n + 2n n n I A I IJ A I B J A I v o o v p C same form as T + higher rank excitation ops. Complicated scheme. Not further pursued.

9 Most favorable scheme: hole-type CI. - Doubly occupy all orbitals that are strongly occupied in the full wave function (" user's selection"). This specifies the vacuum state 0 having 2 N I electrons. - The operator C would consist of nh and (n+1)h-1p type of excitation operators. The number of electrons in the state of interest is 2 N I - n =. N el - Typical example: systems with one low lying virtual orbital increase the number of electrons by two 0. C= c a a + C a a a a IJ I J I< J I< J< K, c IJcK I J c K State Selective DIP-MRCC scheme. In principle applicable to Biradicals Breaking of single bonds Twisting of double bonds Many photo-chemical reactions

10 Possible redundancy in parameterization. Consider 2-electron system, while 0 = 1122 contains 4 electrons (e.g. a stretched H 2 molecule) aaaar a b i j 0 aa a b vac independent of i, j { 1122,,, }. ab We can change the values of the parameters t ij without actually changing the wave function! In other words 0 La i a aa ja b are linearly dependent. Metric matrix S = 0 Lq q L 0 is singular. λµ λ µ In practice we are dealing with a set of non-linear equations that are nearly singular! Typical small 2 3 eigenvalues metric matrix (and smaller). Solution: project out nearly singular component of T. Necessary to obtain convergence. How?, Updating t: solve 0 Lqλ H0 T L 0 = Yλ (residual) Solve linear equation (in direct fashion) for X = H, T 0. Use singular value decomposition to obtain regularized component. At convergence only regular part of residual will vanish.

11 Size consistency Consider non-interacting subsystems A... B Open-shell orbitals i, a E=E A +E B Closed Shell Orbitals j, b The open-shell problem (operator C) should be completely localized on system A. If we assume that H has localized matrix elements then the additional b b condition is Φ j H 0 = h = 0. (for excitations on B). Relevant Matrix structure of H exc. ij ijck ijck ij X X X ijck X X X ijck 0! 0 X j If C is localized on A then equations reduce to CCSD for system B. 0 LQ 2HC 0 0Q 2H 0 = 0 where Q 2 indicates excitation on system B. Results are size-consistent provided...

12 Equations for single excitations are crucial: Possibilities: c c 1. Φ K H 0 = hk = 0?? Typically leads to large singles and poor results. 0 is not Hartree-Fock determinant for N+2 electrons La Kab HC 0 = 0?? These equations are completely redundant (!) as 0 La K a b is 3h1p excitation. T 1 = 0 would be solution. Violates size-consistency (because ill defined) LaKab HL 0 = 0 Good, size-consistent results. Small singles coefficients. Formulation of n-hole type MCSCF: 0 T 1 T 1 La a e He L 0 = 0 K b Solve eqns. in Brueckner fashion and use T 1 as orbital rotation parameters variationally optimal 0 that we use in practice. Closed-shell orbital optimization.

13 Algorithm to solve MRCC equations. 1. Solve Brueckner MCSCF equations L 0 Variant A. Full MRCC scheme: iterate a. Calculate relevant parts of H = e T He T b. Solve IP-/ EA-/ DIP-EOM-CC eqns C c. Factorize (approximately) C SC 2 0 = 1 0 S d. Calculate residuals 0 LQ 2H ( 1+ S ) C 1 0 and 0 LQ 1HC 1 0 (Coded by APG) e. Find regularized updates for T, T 1 2 Variant B. State-Selective (ST)EOM-CC Iteratively solve 0 LQ 2HL 0 = 0 LQ 1HL 0 =0 (Coded by APG) Calculate H = e T He T Solve IP-/ EA-/ DIP-EOMCC or STEOM equations.

14 Illustrations Redundancy / Projection. Consider H 2 molecule at stretched geometry 2 2 Ψ cgσ g + cuσu At R=1.5 Å in TZ2P basis c = , c = Various exact solutions to the problem using different projections 0 L Projection used t( σ g πu) t( σu πu) 2 2 c σ + c σ g g u u 2 σ g σ u g u 4 4 O 2 : Vacuum state in MRCC is 0 =...π u π g. Low lying excited states π 2 g but mixing with π 2 u. Results depend slightly upon projection used: Method 3 Σ u, 1 g 1 Σ g,+ R e (Å) ω (cm -1 ) E (a.u.) (ev) (ev) MRDIP t-mrdip CCSD CCSD(T)

15 Solving for the t-amplitudes: In every iteration calculate residuals Q Solve 0 Re H, TR 0 = Q ij ab 0 ij ab i a, Q - Equations can be (nearly) singular - Not fully dominant on diagonal special singular value decomposition solver. ij ab What is H0 = ε p p p? p Amplitudes t ab ij correspond to excitation from (partially) occupied orbitals into virtual orbitals: - ε i : ionization potentials obtained using extended Koopmans' Theorem (EKT). - Diagonalize ( H E 0 ) over states ir 0 ε i : Completely different (~ 20 ev) from diagonalizing Fock matrix! - ε a : Electron affinities corresponding to EKT. - Diagonalize ( H E 0 ) over states a R 0 ε a. : Equivalent to diagonalizing Fock matrix.

16 Is the Algorithm numerically stable?? Severe test: Calculate vibrational frequencies of twisted ethylene in lowest singlet and triplet state. H H H H Symmetry of molecule and character of orbitals changes as finite displacements are made to calculate Hessian. Triplet State Singlet State CCSD MRCC STEOM MRCC STEOM R CC R CH < HCC ω 1 (E) ω 2 (B 1 ) i 1756 i ω 3 (E) ω 4 (A 1 ) ω 5 (A 1 ) ω 6 (B 2 ) ω 7 (A 1 ) ω 8 (B 2 ) ω 9 (E)

17 Accuracy of MRCC, SS-EOMCC and STEOM compared to CCSD and CCSD(T). Consider singlet and triplet in ethylene at D 2h and D 2d geometries (barrier to bond rotation). MRCC SS-DIPEOM (1.24 * ) DIPEOM SS-STEOM STEOM CCSD CCSD(T) Method D 2d D 2d Barrier D 2h triplet ST-spitting D2d(T)- ST-splitting total E D2h(S) (a.u.) (kcal/mol) (kcal/mol) (ev) Method D 2d D 2d Barrier D 2h triplet ST-spitting D2d(T)- ST-splitting total E D2h(S) (a.u.) (kcal/mol) (kcal/mol) (ev) DIP-STEOM [2-] DIP-EOM [2-] SS-DIPEOM [mb] SS-DIPEOM [R] CCSD CCSD[T]

18 - For triplet states (spin-adapted) MRCC results are virtually identical to ROHF-CCSD. - MRCC is well balanced: both barrier and ST-splitting at D 2h geometry compare well to CCSD(T). - STEOM and DIPEOM energetics not very accurate for larger basis sets due to use of di-anion orbitals in the reference state.

19 The Benzynes p-benzyne m-benzyne o-benzyne Method para-benzyne meta-benzyne ortho-benzyne T S-T T - T(p) S-T T - T(p) S-T total E (a.u.) (kcal) (kcal) (kcal) (kcal) (kcal ) MRCC B-MRCC SS-DIPEOM STEOM CCSD (2.96) 5.8 (6.5) CCSD(T) (3.72) 16.5 (17.8) In general good agreement between MRCC based on MCSCF orbitals and full Brueckner MRCC. - Meta-benzyne is difficult system, e.g. large difference between ROHF- and UHF-CCSD(T). - Poor results for SS-DIPEOM : large difference between relaxed C 1 and MCSCF R 1.

20 S-T splitting in Methylene CH 2 TZ2P basis, MBPT[2] geometries. MRCC SS-DIPEOM STEOM CCSD CCSD(T)

21 Comparisons with Full CI results. H 4 ground state H 4 First Excited State Geom. FCI B-MRCC B-SS- B-MRCC CCSDT DIPEOM relaxed Total E (H) E in mh Geom. FCI B- MRCC B-SS- DIPEOM B-MRCC relaxed Total E (H) E in mh H 8 ground and excited state FCI B-MRCC B-SS- DIPEOM B-MRCC Relaxed STEOM Ground State ( E in mh) First Excited State ( E in mh)

22 Computer Aided Development of Electronic Structure Programs Dr. Victor Lotrich and M. Nooijen Why? - New tools needed to tackle resilient problems in quantum chemistry. - Such tools tend to be highly complex, but are not necessarily computationally expensive. - The 'right' tools are not known yet. Experimentation is necessary. - Current strategies: prone to errors, tedious, labor intensive, repetitive, not pedagogical for students. - Efficiency, robustness and applicability of programs can be increased.

23 The Automatic Program Generator (APG) - Simple input to specify equation. - Evaluation of Wick's theorem to write operator product in normal order. - Additional manipulations if needed to derive desired amplitude equations (e.g. take derivatives, multiply by density matrices). - Factorization of equations. Define intermediates to multiply quantities efficiently. - General set of library routines to perform elementary operations (addition and multiplication of up to 6-index quantities). - Automated computer implementation of equations. The APG can provide an efficient FORTRAN implementation for a wide variety of many-body methods.

24 Automated Derivation of Many-Body Equations. Generally: Apply Wick's theorem to a product of (normal ordered) operators Determine R = ABCD... A Apq..; rs.. p rq s..... = n s R= R + R p q + R p qr s +... n s n s 0 pq pqrs Rpq; rs = AstuvBqurvCptxyDxy,... etc. - Apply Wick's theorem - Collect unique terms - Spin-integration In our MRCC applications we typically get spin-adapted terms (Feynman diagrams) that consist of multiplications of 2-4 operators each!

25 Factorization Use only 2 elementary operations to sum all terms. Addition: Apqrs = Bpqrs + Cpqrs Matrix-Multiplication. A = B + C D pq; rs pq; rs pq; tu tu; rs Split up general term R = A B C D n pq; rs stuv qurv ptxy xy 1 6 st; qr stuv qurv 2 4 pt pt; xy xy pq; rs st; qr pt I = A B n I = C D n R = I I n Intermediates need to be stored and might be (partially) reused to evaluate other terms. Definition of intermediates effects operation count and storage costs. Optimal factorization consitutes n-p complete problem in mathematics. Typically there may be (!) different ways to factorize. 9

26 Automation of implementation. Efficient and flexible implementation of each elementary operation. Add operators: A( pq, rs) = β B( pq, rs) + γ C( pq, rs) Multiply operators A( pq.., rs..)* B( rs.., tu..) + C( pq.., tu..) Library of hand written subroutines diagram222, 242,...444,..264, 446, 266,...,466 - Reorder/select quantities according to indices. - Multiply (efficient, vectorized). - Reorder result to final form. - Process everything in manageable chunks. - Simple arguments. - Symmetry blocked algorithm. Final computer code transparent series of 'diagram' calls. Final code is written by computer itself, without any loss of efficiency.

27 Example: Evaluating DIP-MRCC equations: ij ab ab ij T T Eq = 0 Le e He ( 1+ S ) R 0 ij ab i a j b i a j b i a j b i a j b e = aaaa + aaaa + aaaa + aaaa define cd Z = z e klcd,,, kl T T G Ze He ( S ) p g g e r 1 pq r p grs e rs = 1+ = pq Evaluate G by Wick's Theorem. i j ij kl F = 0 LGR 0 = g0 + gjdi + gkl Dij i i Dj = 0 Le j R 0 : spin-free density matrices. (Explicit formulas are worked out by hand) kl cd Task for APG - Evaluate specific G matrix elements - Multiply by density matrices F - Take derivative w.r.t. z ab ab F ij ; Eqij = ab Zij - Factorize equations and write program. Instant Fortan in just 15 minutes!

28 Summary: - State-Selective Equation-of-Motion Coupled Cluster. - Transform then Diagonalize approach to general openshell problems. - Equations for transformation coefficients t fairly insentitive to details. - Suitable zeroth-order Hamiltonian using "Extended Koopmans Theorem. + Perturbation Theory. - General Equations with o 2 v 4 scaling.!! Computer Aided Implementations!! Work in Progress: "Tensor Contraction Engine" Computer generated implementations. Optimize traffic of data flow Cache Ram Disk Recomputation. Sadayappan, Baumgartner, Harrison, Hirata, Corciova, Bernholdt, Pitzer, Nooijen