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PhD Defence Part 2
- 1. Second Order Generalized Van Vleck Perturbation Theory Molecular Gradients and Nonadiabatic CouplingTerms Daniel P. Theis University of North Dakota Chemistry Department Grand Forks, ND
- 2. 2 • Introduction Outline Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. • GVVPT2 Gradiengts • GVVPT2 Nonadiabatic Coupling Terms
- 3. 3 1. GVVPT2 is a post-MCSCF, perturbation based electronic structure method. Benefits of the GVVPT2 Method Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. 2. GVVPT2 can determine accurate electronic energies for systems with several low lying, nearly degenerate electronic states. 3. GVVPT2 potential energy surfaces are continuous, differentiable functions of the geometry, that ensure the evaluation of molecular gradients. 4. A highly efficient algorithm has been constructed for the GVVPT2 method, which combines the advantages of the macroconfiguration and GUGA techniques.
- 4. 4 Λ ∈ Λ ∑=Ψ t Lt t MRCI AF T HMM HQM HMQ HQQ Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. MCSCF The GVVPT Methodology ◦ Relatively Small CSF space, LM ◦ Neglects Dynamic Correlation (LQ) mI Lm m MC I CF M ∑∈ =Ψ ◦ Large CSF Space, LT = LM ⊕ LQ ◦ Includes Dynamic Correlation. MRCISD ( ) 0=−∑∈ ′′ MLm mImm MC Imm CEH δ ( ) 0=−∑∈ Λ ′Λ′ TLt ttt MRCI tt AEH δ
- 5. 5 ∑∈ Λ Λ =Φ MLm mm GV AF HQM HMQ Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. eff MMH GVVPT Methods The GVVPT Methodology HQQ ,ˆ GVVGVV ΛΛ ΦΩ=Ψ ( ) ,0=−∑∈ Λ ′Λ′ MLm mmm GVVeff mm AEH δ mm eff mm FHFH ′ + ΩΩ= ˆˆˆ ◦ Heff is the same dimension as HMM ◦ Perturbative corrections from HQM, HMQ, and the diagonal elements of HQQ add dynamic correlation.
- 6. 6 eff QPH eff PQH eff PSH eff SPH eff PPH eff SSH eff SQH eff SQH eff QQH The GVVPT Methodology Shavitt, I.; Redmon, L. T. J. Chem. Phys. 1980, 73, 5711. Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351. Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. The effective Hamiltonian is formed by: • LM = LP ⊕ LS.
- 7. 7 eff QPH eff PQH eff PSH eff SPH eff PPH eff SSH eff SQH eff SQH eff QQH ( ) 0HH XX == −− QP eff QP ee The GVVPT Methodology Shavitt, I.; Redmon, L. T. J. Chem. Phys. 1980, 73, 5711. Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351. Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. X e ˆ ˆ =Ω QPQPPQPQX FXFFXF + −=ˆ The effective Hamiltonian is formed by: • LM = LP ⊕ LS. • Define Ω, X, and XQP such that:ˆ ˆ ◦ ◦ ◦
- 8. 8 eff QPH eff PQH eff PSH eff SPH eff PPH eff SSH eff SQH eff SQH eff QQH • Perturbatively expand Heff and X. ( ) 0HH XX == −− QP eff QP ee The GVVPT Methodology Shavitt, I.; Redmon, L. T. J. Chem. Phys. 1980, 73, 5711. Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351. Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. X e ˆ ˆ =Ω QPQPPQPQX FXFFXF + −=ˆ The effective Hamiltonian is formed by: • LM = LP ⊕ LS. • Define Ω, X, and XQP such that:ˆ ˆ ◦ ◦ ◦ { }( )+ ++= )2()2( 2 1)2( MMMMMM eff MM ZZHH ( ) PMQPMQPMMPMMM CXHCCIZ )1()2( 2 −= GVVPT2 Effective Hamiltonian
- 9. 9 Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. The Origen of GVVPT2’s Stable Potential Energy Curves )1( qIX I emε∆ a 2a 3a 4a -4a -3a -2a -a a = HqI I Iq II qI qI ee HH X mm εεε ∆ −= − −≈ 0 )1( ( ) QIQ MC IQQQI E HIHX 1)1( − −= Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351.
- 10. 10 Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. )1( qIX I emε∆ a 2a 3a 4a -4a -3a -2a -a a = HqI I Iq qI e E H X m∆ −≈)1( ( ) ( )∑∈ +∆+∆=∆ e eee Lq qI III HE m xxxx mmm 22 4 1 2 1 )()()()( εε◦ ( ) QIQ MC IQQQI E HIHX 1)1( − −= Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351. The Origen of GVVPT2’s Stable Potential Energy Curves
- 11. 11 qI I qI HDX em≈)1( Hoffmann, M. R. et. al. J. Chem. Phys. 2002, 117, 4133. ( ) I I I e e e E E D m m m ∆ ∆ −= tanh )1( qIX I emε∆ a 2a 3a 4a -4a -3a -2a -a a = HqI ◦ ( ) QIQ MC IQQQI E HIHX 1)1( − −= Kuhler, K.; Hoffmann, M. R. J. Math. Chem. 1996, 20, 351. The Origen of GVVPT2’s Stable Potential Energy Curves
- 12. 12 where each λj is determine from … e1(x;λ(x)) = 0 e2(x;λ(x)) = 0EΛ = EΛ(x;λ(x)) Determining Energy Gradients Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.
- 13. 13 ( ) ∑ ∂ ∂ ∂ ∂ + ∂ ∂ = ΛΛΛ j a j jaa x E x E dx dE λ λ )(; xλx where each λj is determine from … e1(x;λ(x)) = 0 e2(x;λ(x)) = 0EΛ = EΛ(x;λ(x)) ( ) 0 )(; = ∂ ∂ ∂ ∂ + ∂ ∂ = ∑j a j j i a i a i x e x e dx de λ λ xλx a j x∂ ∂λ where each is determined from Determining Energy Gradients Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.
- 14. 14 ( ) ( ) ( )∑+= ΛΛ i ii eEL )(;)()(;)(),(; xλxxxλxxξxλx ξ 0= ∂ ∂ Λ j L λ ( ) 0)(; == ∂ ∂ Λ xλxi i e L ξ where the values for ξi and λj are determined by requiring and The Lagrangian Approach of Determining Energy Gradients Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.
- 15. 15 The Lagrangian Approach of Determining Energy Gradients ( ) ( ) ( )∑+= ΛΛ i ii eEL )(;)()(;)(),(; xλxxxλxxξxλx ξ 0= ∂ ∂ Λ j L λ ( ) 0)(; == ∂ ∂ Λ xλxi i e L ξ where the values for ξi and λj are determined by requiring and Under these conditions ( ) aaa x L dx dL dx dE ∂ ∂ == ΛΛΛ )(; xλx Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.
- 16. 16 The Lagrangian Approach of Determining Energy Gradients ( ) ( ) ( )∑+= ΛΛ i ii eEL )(;)()(;)(),(; xλxxxλxxξxλx ξ 0= ∂ ∂ Λ j L λ ( ) 0)(; == ∂ ∂ Λ xλxi i e L ξ where the values for ξi and λj are determined by requiring and Under these conditions ( ) aaa x L dx dL dx dE ∂ ∂ == ΛΛΛ )(; xλx Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111. ∂ ∂ ∂ ∂ ∂ ∂ −= ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Λ Λ Λ n n n n nn n n E E E eee eee eee λ λ λ ξ ξ ξ λλλ λλλ λλλ 2 1 0 01 01 21 22 2 2 1 11 2 1 1 )( )( )( x x x
- 17. 17 ( )∑∈ ′ Λ ′ Λ ′Λ = MLm MP rveff mmmm GVV HAAE )(,)(,)(;)()()( xCxΔxΔxxxx• Electron Structure Parameters that are used in the GVVPT2 Calculations
- 18. 18 ( )∑∈ ′ Λ ′ Λ ′Λ = MLm MP rveff mmmm GVV HAAE )(,)(,)(;)()()( xCxΔxΔxxxx• Electron Structure Parameters that are used in the GVVPT2 Calculations ( ) ( )∑∈ =Φ MLm mI rv mMP rvMC I CF )()(,)(;)(,)(,)(; xxΔxΔxxCxΔxΔxCmI(x) ~•
- 19. 19 ( )∑∈ ′ Λ ′ Λ ′Λ = MLm MP rveff mmmm GVV HAAE )(,)(,)(;)()()( xCxΔxΔxxxx• Electron Structure Parameters that are used in the GVVPT2 Calculations ( ) ( )∑∈ =Φ MLm mI rv mMP rvMC I CF )()(,)(;)(,)(,)(; xxΔxΔxxCxΔxΔxCmI(x) ~• ~)(and)( xx ab r ij v ∆∆ ( ) ( ){ }∑∈ = iGk ki omo k rvmo i )(exp)()(,)(; xΔxxΔxΔx ϕφ v r ( ){ }∑ ∑≠ ∈ + ik kGG Gk ki omo k )(exp)( xΔxϕ • • i∈GiNote:
- 20. 20 ( )∑∈ ′ Λ ′ Λ ′Λ = MLm MP rveff mmmm GVV HAAE )(,)(,)(;)()()( xCxΔxΔxxxx• Electron Structure Parameters that are used in the GVVPT2 Calculations ( ) ( )∑∈ =Φ MLm mI rv mMP rvMC I CF )()(,)(;)(,)(,)(; xxΔxΔxxCxΔxΔxCmI(x) ~• ~)(and)( xx ab r ij v ∆∆ ( ) ( ){ }∑∈ = iGk ki omo k rvmo i )(exp)()(,)(; xΔxxΔxΔx ϕφ v r ( ){ }∑ ∑≠ ∈ + ik kGG Gk ki omo k )(exp)( xΔxϕ • • i∈GiNote: ∑∑ += OMO ijkl b omo ijklA omo ijkl OMO ij B omo ijA omo ijAB FeFgFEFhH ~ ˆ ~ )( ~ˆ~ )()( 2 1 xxx
- 21. 21 ~ ( ) 1)( 2 =∑∈ MLm mIC x [ ] 0)( =−∑∈′ ′′′ MLm Im MC Immmm CEH xδand The Constraint Equations that are used by the GVVPT2 and MRCI Calculations CmI(x)•
- 22. 22 ( ) 0)()(1)( 2 = −+−= ∑ ∑∈′ ′ ∈′′ ′′′′ M M Lm Im Lm Im MC ImmmmmI CCEHe xxx δCmI(x)• ~ The Constraint Equations that are used by the GVVPT2 and MRCI Calculations
- 23. 23 ( ) 0)()(1)( 2 = −+−= ∑ ∑∈′ ′ ∈′′ ′′′′ M M Lm Im Lm Im MC ImmmmmI CCEHe xxx δCmI(x)• ~)(xij v ∆•• ~ [ ] 0)(ˆˆ),()()( =Φ−Φ= ∑ xxxx MC Ijiij MC I N I Iij EEHwe P The Constraint Equations that are used by the GVVPT2 and MRCI Calculations
- 24. 24 ( ) 0)()(1)( 2 = −+−= ∑ ∑∈′ ′ ∈′′ ′′′′ M M Lm Im Lm Im MC ImmmmmI CCEHe xxx δCmI(x)• ~)(xij v ∆•• ~ The Constraint Equations that are used by the GVVPT2 and MRCI Calculations )(xab r ∆ ~ [ ] 0)()()()()( 2 1 =−+== ∑ OCC kl akblabklklababab gghfe xxxxx γ• [ ] 0)(ˆˆ),()()( =Φ−Φ= ∑ xxxx MC Ijiij MC I N I Iij EEHwe P
- 25. 25 ( )∑ ∑∑∑ ×ΦΦ+Γ+= ∂ ∂ Λ Λ PN I MO i aI ii GVV iI MC I GVV MO ij a ijkl GVV ijkl MO ij a ij GVV ij a GVV fPgh x E 2 1 γ• ∑ ∑ ∑∑∑∑ ∆∂ ∂ ×ΦΦ+ Γ+− Γ+= ∆∂ ∂ Λ Λ PN I MO k ij v I kkGVV kI MC I GVV MO xyz jxyz GVV ixyz MO x jx GVV ix MO xyz ixyz GVV jxyz MO x ix GVV jx ij v GVV f P ghgh E 2 1 2222 γγ Efficiently Evaluating the Partial Derivatives of E Λ GVV •
- 26. 26 ( )∑ ∑∑∑ ×ΦΦ+Γ+= ∂ ∂ Λ Λ PN I MO i aI ii GVV iI MC I GVV MO ij a ijkl GVV ijkl MO ij a ij GVV ij a GVV fPgh x E 2 1 γ• ∑ ∑ ∑∑∑∑ ∆∂ ∂ ×ΦΦ+ Γ+− Γ+= ∆∂ ∂ Λ Λ PN I MO k ij v I kkGVV kI MC I GVV MO xyz jxyz GVV ixyz MO x jx GVV ix MO xyz ixyz GVV jxyz MO x ix GVV jx ij v GVV f P ghgh E 2 1 2222 γγ ( ) ( ) ( ) ( ) ( ){ }[ ]MIPMMIPM MC P GVV MIMm Pm MC P GVV Q mI IQMC I GVV mI GVV A CC E CHXHXCΦAHX HXΦHβ X ++ Λ Λ+Λ + Λ Λ + Λ Λ +Φ−+ Φ− ∂ ∂ ×ΦΦ= ∂ ∂ 2 1 2 Efficiently Evaluating the Partial Derivatives of E Λ GVV • •
- 27. 27 ( )∑ ∑ ∑ ∑∈ ∈ ∈ +×= Qe e M P e L q Lm N I qjiijmMIQMqmI GVV ij FEEFW m m m CH ˆ 2 1 γ • Efficiently Evaluating the Partial Derivatives of E Λ GVV ( )∑ ∑ ∑ ∑∈ ∈ ∈ +++×=Γ Qe e M P e L q Lm N I qjilkjiklijlkijklmMIQMqmI GVV ijkl FeeeeFW m m m CH ˆˆ8 1 ( ) ( )∑ ∑∈ ∈ ×= Qe e e L q mqMIQMqImI HX m m m CHHX ( ) ( )∑ ∑∈ ∈ Λ + ×= ∂ ∂ Qe e e L q mqMIQMqIQ mI IQ HV C m m m CHHβ X GVVPT2 Energies – Slowest Step GVVPT2 v ∆ij, r ∆ab, and CmI Derivatives – Slowest Steps • • • ~ 2 × time of (HX)mI ~ 1 × time of (HX)mI
- 28. 28 Molecule Analytical Values Deviation from Numerical Values Geometry Description X Y Z X Y Z H2 CO C -0.171726 0.205990 0.000000 -6.90×10-7 -8.00×10-8 0.00 CH1 Str. (+0.5 Å) O 0.081525 -0.025708 0.000000 3.70×10-7 4.20×10-7 0.00 H1 0.240019 -0.424267 0.000000 -6.00×10-8 7.00×10-8 0.00 H2 -0.149818 0.243985 0.000000 9.00×10-8 -1.20×10-7 0.00 LiH (X 1 Σ+ ) H 0.000000 0.000000 -0.014113 0.00 0.00 0.00 Avoided Crossing Li 0.000000 0.000000 0.014113 0.00 0.00 0.00 LiH (A 1 Σ+ ) H 0.000000 0.000000 -0.004441 0.00 0.00 0.00 Avoided Crossing Li 0.000000 0.000000 0.004441 0.00 0.00 0.00 • cc-pVTZ Basis Set Technical Details: H2CO (Cs – Broken Sym.) • RCO = 1.205 Å, RCH = 1.611 Å, and RCH = 1.111 Å. • 2 Val. Groups {612 40 ; 611 41 ; 610 42 } ∀ ∠HCH = 116.1o and ∠OCH = 121.9o 1 2 • Roos Aug. TZ Basis Set Technical Details: LiH (C∞v) • 9:1 SA-MCSCF MOs • (2:10)-CAS + 1 Core Orb. • RLiH = 3.400 Å Analytical GVVPT2 Gradients for H2CO and LiH
- 29. 29 The Lagrangian Approach of Determining Nonadiabatic Coupling Terms ( ) ( ) ( )∑+= Λ′ΛΛ′Λ i ii eg )(;)()(;)(),(; xλxxxλxxξxλx ξL where gΛΛ'(x) is a function that makes 0= ∂ ∂ Λ′Λ jλ L andSetting 0= ∂ ∂ Λ′Λ iξ L , makes: aaaaa xdx d dx dg dx d dx d ∂ ∂ === Ψ Ψ− Ψ Ψ Λ′ΛΛ′ΛΛ′ΛΛ Λ′ Λ′ Λ )()()()()( 2 1 xxxxx LL Ψ Ψ− Ψ Ψ= Λ Λ′ Λ′ Λ Λ′Λ aaa dx d dx d dx dg )()()( 2 1 xxx
- 30. 30 The Lagrangian Approach of Determining Nonadiabatic Coupling Terms ( ) ( ) ( )∑+= Λ′ΛΛ′Λ i ii eg )(;)()(;)(),(; xλxxxλxxξxλx ξL where gΛΛ'(x) is a function that makes ( ) ( ){ } Λ′ ΛΛ′ ++Λ Λ′Λ − ++Θ−Θ= M eff MMGVGVMMPMMPPMMPM GV EE CCg AxHxFFxxAx )( 1 )()()()( 2 1 Ψ Ψ− Ψ Ψ= Λ Λ′ Λ′ Λ Λ′Λ aaa dx d dx d dx dg )()()( 2 1 xxx For GVVPT2
- 31. 31 The Lagrangian Approach of Determining Nonadiabatic Coupling Terms ( ) ( ) ( )∑+= Λ′ΛΛ′Λ i ii eg )(;)()(;)(),(; xλxxxλxxξxλx ξL where gΛΛ'(x) is a function that makes ( ) ( ){ } Λ′ ΛΛ′ ++Λ Λ′Λ − ++Θ−Θ= M eff MMGVGVMMPMMPPMMPM GV EE CCg AxHxFFxxAx )( 1 )()()()( 2 1 Ψ Ψ− Ψ Ψ= Λ Λ′ Λ′ Λ Λ′Λ aaa dx d dx d dx dg )()()( 2 1 xxx For GVVPT2 )()()(ˆ)( )2()2( xAxFxx Λ Λ Ω=Ψ MM GV
- 32. • Geometry optimizations, gradients calculations, and frequency calculations verify that the GVVPT2 method accurately describes the chemically important regions of most potential energy surfaces. Conclusions 32 • The GVVPT2 gradients are continuous across potential energy surfaces, including regions of avoided crossings. • Analytic formulas for GVVPT2 molecular gradients and nonadiabatic coupling terms have been developed which scale at approximately 2-3 times the speed of the GVVPT2 energy. • Computational implementation of GVVPT2 analytic gradients show excellent agreement with finite difference calculations.
- 33. Dr. Mark R. Hoffmann Dr. Yuriy G. Khait Patrick Tamukang Rashel Mokambe Jason Hicks Erik Timmian Dr. Theresa Windus Dr. Klaus Ruedenberg Acknowledgements 33