Crittenden DL

Sibaev M

UC Research Repository

Introduction Electronic structure PES construction PES validation OutlookA transformative approach for constructinganharmonic force fieldsMarat Sibaev and Deborah Crittenden∗University of Canterbury, Christchurch, New Zealand∗deborah.crittenden@canterbury.ac.nzIntroduction Electronic structure PES construction PES validation OutlookPrimary goalRoutine, black-box calculation of anharmonic vibrationalfrequencies for semi-rigid molecules, extensible to selectedvibrational modes of larger systems (clusters, materials).Introduction Electronic structure PES construction PES validation OutlookComputational considerationsOverall trade-offs between accuracy and computational cost:Introduction Electronic structure PES construction PES validation OutlookElectronic structure methods for vibrational frequenciesWeak coupling between electronic and nuclear degrees offreedom in wavefunctionHarmonic contribution to fundamentals >> anharmoniccontribution⇓Use harmonic frequencies for benchmarking the performance ofdifferent electronic structure methodsIntroduction Electronic structure PES construction PES validation OutlookElectronic structure methods for vibrational frequenciesBenchmark data setSpectroscopically accurate potential energy surfaces for 50small molecules (3-6 atoms) implemented from the literatureGround and excited statesAnionic, cationic and unchargedOrganic and inorganicAt least CCSD(T)/aug-cc-pVQZ qualityExpanded in symmetry-adapted internal coordinatesAnharmonic fundamentals within 5 cm−1 of experimentHarmonic frequencies calculated from anharmonic PESIntroduction Electronic structure PES construction PES validation OutlookElectronic structure methods for vibrational frequenciesApproximate electronic structure methodsB3LYP/6-31G(d,p), B3PW91/6-31G(d,p), PBE0/6-311G(d,p)EDF2/(aug-)cc-pVTZM05/6-311+G(2df,2p), M06/6-311+G(2df,2p)MP2/(aug-)cc-pVTZ* All DFT calculations performed using ultra-fine grids and tightSCF convergence thresholds.Introduction Electronic structure PES construction PES validation OutlookElectronic structure methods for vibrational frequenciesErrors (cm−1)Method Mean MaxB3LYP 28 223B3PW91 25 245PBE0 27 228EDF 27 238M05 36 204M06 32 178MP2 23 128Mean anharmonicity = 47 cm−1Max. anharmonicity = 237 cm−1Introduction Electronic structure PES construction PES validation OutlookElectronic structure methods for vibrational frequenciesErrors (cm−1)Method Mean MaxB3LYP 28 223B3PW91 25 245PBE0 27 228EDF 27 238M05 36 204M06 32 178MP2 23 128Mean anharmonicity = 47 cm−1Max. anharmonicity = 237 cm−1Introduction Electronic structure PES construction PES validation OutlookElectronic structure methods for vibrational frequenciesErrors (cm−1)Method Mean MaxB3LYP 28 223B3PW91 25 245PBE0 27 228EDF 27 238M05 36 204M06 32 178MP2 23 128Mean anharmonicity = 47 cm−1Max. anharmonicity = 237 cm−1Introduction Electronic structure PES construction PES validation OutlookElectronic structure methods for vibrational frequenciesTake-home messagesDon’t use DFT for anharmonic frequency calculationsDon’t even use MP2Coupled-cluster models with at least at TZ basis requiredMay be able to get away with a lower level of theory for theanharmonic part of the potential?If you can only afford a poor electronic structure model, youmay as well use a poor anharmonicity model∗∗M. Sibaev & D. L. Crittenden, J. Phys. Chem. A, 2015, doi:10.1021/acs.jpca.5b11386Introduction Electronic structure PES construction PES validation OutlookThe importance of coordinate choiceDescribing the potential energy surfaceForce field coordinates required for concise PES expansionCurvilinear internal coordinates - bond lengths, bond angles,dihedral angle, out-of-plane anglesRedundant setSolving the nuclear Schrödinger equationNormal mode coordinates diagonalize kinetic energy operatorRectilinear coordinates - Cartesian displacement vectorsNon-redundant setIntroduction Electronic structure PES construction PES validation OutlookYou can’t have your cake and eat it tooTwo possible solutions:Transform coordinates in which PES is expandedlow order expansion in internal coordinates → higher orderexpansions in Cartesian normal mode coordinatesdescribes curvature of potential energy surfaceTransform coordinates in which kinetic energy operator is expressedexpress KE operator in same non-redundant set of internalcoordinates as PESdescribes curvature of internal coordinate setIntroduction Electronic structure PES construction PES validation OutlookYou can’t have your cake and eat it tooTwo possible solutions:Transform coordinates in which PES is expandedlow order expansion in internal coordinates → higher orderexpansions in Cartesian normal mode coordinatesdescribes curvature of potential energy surfaceTransform coordinates in which kinetic energy operator is expressedexpress KE operator in same non-redundant set of internalcoordinates as PESdescribes curvature of internal coordinate setIntroduction Electronic structure PES construction PES validation OutlookPES expansions for HCl1.0 1.5 2.0 2.5r (Å)050001000015000200002500030000Vn(r)(cm−1)1.0 1.5 2.0 2.5r (Å)050001000015000200002500030000Vn(r−rer )(cm−1)Introduction Electronic structure PES construction PES validation OutlookFundamental vibrational frequency of HCl (cm−1)n Vn(r − re) Vn( r−rer )2 2990.9 2927.83 2730.5 2911.84 2901.6 2886.15 2881.8 2885.26 2885.6 2885.2∞ 2885.2 2885.2Introduction Electronic structure PES construction PES validation OutlookFundamental vibrational frequency of HCl (cm−1)n Vn(r − re) Vn( r−rer )2 2990.9 2927.83 2730.5 2911.84 2901.6 2886.15 2881.8 2885.26 2885.6 2885.2∞ 2885.2 2885.2Introduction Electronic structure PES construction PES validation OutlookFundamental vibrational frequency of HCl (cm−1)n Vn(r − re) Vn( r−rer )2 2990.9 2927.83 2730.5 2911.84 2901.6 2886.15 2881.8 2885.26 2885.6 2885.2∞ 2885.2 2885.2Introduction Electronic structure PES construction PES validation OutlookFundamental vibrational frequency of HCl (cm−1)n Vn(r − re) Vn( r−rer )2 2990.9 2927.83 2730.5 2911.84 2901.6 2886.15 2881.8 2885.26 2885.6 2885.2∞ 2885.2 2885.2Introduction Electronic structure PES construction PES validation OutlookMoving on to molecules1 Define redundant internal coordinate set, ZrSimon-Parr-Finlan for stretchesPolynomial for bond angles, torsion angles, out-of-plane angles2 Construct non-redundant internal coordinate set, ZCan use symmetry-adapted internal coordinates for simple,small, symmetrical moleculesOtherwise, how best to define appropriate linear combinations?3 Generate concise PES in internal coordinates, Vn(Z)4 Transform into higher order expansion in normal modecoordinates, e.g. V6(Q)5 Solve nuclear vibrational Schrödinger equationIntroduction Electronic structure PES construction PES validation OutlookDefining non-redundant internal coordinatesThe simplest approachUse Wilson G matrix method:G = BrBrT, Nint × Nint matrixBr =dZrdX , Nint × 3Natom matrixdiagonalize GU = Nmode × Nint transformation matrix comprising Nmodeeigenvectors with non-zero eigenvaluesZ = UZrCalculate energy derivatives by numerical differentiation:with respect to displacements along Zianalytic second derivatives derived from ab initio Hessian databut what step sizes to take along each coordinate, Zi?and what order of expansion will be required?Introduction Electronic structure PES construction PES validation OutlookDefining non-redundant internal coordinatesAn extra stepDefine curvilinear normal mode coordinates, Q̃, as a linearcombination of non-redundant internal coordinates, Z, suchthat dQ̃dX =dQdXL = dXdQ transforms between Cartesian and normal modecoordinates, derived from normal mode analysisL̃ = dXdQ̃= dXdQ = L, by definitionB = dZdX → Wilson B-matrix for non-redundant coordinate setQ̃ = (BL̃)−1ZCalculate energy derivatives by numerical differentiation:with respect to displacements along Q̃ianalytic second derivatives derived from ab initio Hessian datastep size inversely proportional to frequencycan tailor order of expansion to character of vibrational modeIntroduction Electronic structure PES construction PES validation OutlookPES expansionsConcise expansion in curvilinear normal mode coordinatesV4(Q̃) =12!∑i,jF̃i,j Q̃i Q̃j +13!∑i,j,kF̃i,j,kQ̃i Q̃j Q̃k +14!∑i,j,k,lF̃i,j,k,l Q̃i Q̃j Q̃kQ̃l⇓ Coordinate transformation∗Higher order expansion in rectilinear normal mode coordinatesV6(Q) =12!∑i,jFi,jQiQj +13!∑i,j,kFi,j,kQiQjQk +14!∑i,j,k,lFi,j,k,lQiQjQkQl +15!∑i,j,k,l,mFi,j,k,l,mQiQjQkQlQm +16!∑i,j,k,l,m,nFi,j,k,l,m,nQiQjQkQlQmQn∗W. D. Allen, A. G. Császár, V. Szalay, I. M. Mills, Mol. Phys., 1996, doi:10.1080/002689796173138Introduction Electronic structure PES construction PES validation OutlookCoordinate transformation procedureF̃i =dVdQ̃i= 0F̃i,j =d2VdQ̃idQ̃jF̃i,j,k =d3VdQ̃idQ̃jdQ̃kF̃i,j,k,l =d4VdQ̃idQ̃jdQ̃kQ̃l⇒Fi =dVdQi= 0Fi,j =d2VdQidQjFi,j,k =d3VdQidQjdQkFi,j,k,l =d4VdQidQjdQkQlFi,j,k,l,m =d5VdQidQjdQkQlQmFi,j,k,l,m,n =d6VdQidQjdQkQlQmQnIntroduction Electronic structure PES construction PES validation OutlookCoordinate transformation procedureShorthand notation:{dVdQ̃}4 dQ̃dQ5======⇒{dVdQ}6Requires:repeated application of chain rule∗derivatives of Q̃ with respect to Q, to 5th order, themselvesformed through a sequence of coordinate transformations:{dZdX}5 dXdQ==⇒{dZdQ}5 dQ̃dZ==⇒{dQ̃dQ}5required transformation matrices defined during normal modeanalysis, construction of Q̃∗W. D. Allen, A. G. Császár, V. Szalay, I. M. Mills, Mol. Phys., 1996, doi:10.1080/002689796173138Introduction Electronic structure PES construction PES validation OutlookCoordinate transformation procedureReduced mode-representation variants:Restrict number of independent indices in F̃ :V3MR4 (Q̃)→ V6(Q)V2MR4 (Q̃)→ V6(Q)Restrict number of independent indices in F :V4(Q̃)→ V5MR6 (Q)V4(Q̃)→ V4MR6 (Q)V4(Q̃)→ V3MR6 (Q)Both:Sensible combinations of the abovem̃ = mode-coupling in curvilinear normal mode potentialm = mode-coupling in rectilinear normal mode potentialIntroduction Electronic structure PES construction PES validation OutlookComputational scalingStep Scaling law Example:O(· · · ) V4(Q̃)⇒ V6(Q)ab initio calculations N7basis Nm̃−2mode N7basis N2modeconstruction ofdQ̃dQNm−1modeNint N5modeNintcoordinate transformation Nm̃+mmode N10modem̃ = mode-coupling in curvilinear normal mode potentialm = mode-coupling in rectilinear normal mode potentialIntroduction Electronic structure PES construction PES validation OutlookImplementation details{dZdX}5matrix elements obtained via:stretching bending torsional out-of-planehard-coded expressions r cos θ sinφ τcoordinate transformation 1− e−α(r−re) θ φ sin τ(r − re)/rsymbolic differentiation other other other otherAll transformation formulae hard-codedEnables general non-linear transformations between anydefined coordinate systemsInterfaces with GAMESS and CFOUR program packagesCode available on sourceforge: PyPES extensible libraryIntroduction Electronic structure PES construction PES validation OutlookPES validationTransformed PES expansions validated by:calculating anharmonic vibrational fundamental frequencies,using VPT2-screened VCI algorithmcomparing results against benchmark literature valuesinternal comparison - choice of internal coordinate system,completeness of internal coordinate PESinternal comparison - convergence with respect to expansionorder, mode-representationIntroduction Electronic structure PES construction PES validation OutlookNuclear vibrational structure methodVPT2-screened VCI1 Construct harmonic oscillator basis states, according tomaximum “excitation level” (total vibrational quanta)2 Choose states for inclusion in VCI matrix based on their VPT2corrections to harmonic frequencies, using loose screeningthreshold3 Diagonalize newly formed sparse VCI matrix4 Choose more states based upon their VPT2 contribution tonew, anharmonic fundamentals, tightening screening threshold5 Repeat steps (3) and (4) until no new states are selected, orfundamental transition energies don’t changeIntroduction Electronic structure PES construction PES validation OutlookEtheneRedundant internal coordinates:5 × bond lengths6 × bond angles4 × dihedral anglesSextic force field in non-redundant set ofsymmetry-adapted internal coordinates,V6(Zref), implemented from the literature11. T. Delahaye et al., J. Chem. Phys., 2014, 141, 104301Introduction Electronic structure PES construction PES validation OutlookCoordinate definition and PES truncation errors (cm−1)0 500 1000 1500 2000 2500 3000 3500ν (cm−1)−30−20−100102030∆ν(cm−1)Reference:V6(Zref)→ V6(Q)Approximation:V4(Q̃)→ V6(Q)Introduction Electronic structure PES construction PES validation OutlookReduced mode representation in internal coordinates0 500 1000 1500 2000 2500 3000 3500ν (cm−1)−30−20−100102030∆ν(cm−1)3MR2MRReference:V4(Q̃)→ V6(Q)Approximations:V3MR4 (Q̃)→ V6(Q)V2MR4 (Q̃)→ V6(Q)Introduction Electronic structure PES construction PES validation OutlookReduced mode representation in normal mode coordinates0 500 1000 1500 2000 2500 3000 3500ν (cm−1)−30−20−100102030∆ν(cm−1)5MR4MR3MRReference:V4(Q̃)→ V6(R)Approximations:V4(Q̃)→ V5MR6 (Q)V4(Q̃)→ V4MR6 (Q)V4(Q̃)→ V3MR6 (Q)Introduction Electronic structure PES construction PES validation OutlookTesting - moleculesA series of internal coordinate force fields and higher ordertransformed normal mode force fields:V4(Q̃)→ V6(Q) (reference)V3MR4 (Q̃)→ V4MR6 (Q)V2MR4 (Q̃)→ V4MR6 (Q)were generated for:4 × 5-atom molecules8 × 6-atom molecules5 × 7-atom molecules6 × 8-atom moleculesIntroduction Electronic structure PES construction PES validation Outlook2MR QFF in internal coordinates0 500 1000 1500 2000 2500 3000 3500 4000ν (cm−1)−40−30−20−10010203040∆ν(cm−1)Reference:V4(Q̃)→ V6(Q)Approximations:V2MR4 (Q̃)→ V4MR6 (Q)Introduction Electronic structure PES construction PES validation Outlook3MR QFF in internal coordinates0 500 1000 1500 2000 2500 3000 3500 4000ν (cm−1)−30−20−100102030∆ν(cm−1)Reference:V4(Q̃)→ V6(Q)Approximations:V3MR4 (Q̃)→ V4MR6 (Q)Introduction Electronic structure PES construction PES validation OutlookAccuracy vs computational costMean Scaling law, O(· · · )Potential Absolute PES CoordinateError (cm−1) construction transformationV2MR4 (Q̃)→ V6(Q) 4.7 N7basis N8modeV3MR4 (Q̃)→ V6(Q) 0.7 N7basisNmode N9modeV4(Q̃)→ V3MR6 (Q) 4.7 N7basisN2mode N7modeV4(Q̃)→ V4MR6 (Q) 0.6 N7basisN2mode N8modeV4(Q̃)→ V5MR6 (Q) 0.03 N7basisN2mode N9modeV3MR4 (Q̃)→ V4MR6 (Q) 1.0 N7basisNmode N7modeIntroduction Electronic structure PES construction PES validation OutlookSummaryV3MR4 (Q̃)→ V4MR6 (Q) PES givesmaximum error < 12 cm−1 and average error < 1 cm−1relative to V4(Q̃)→ V6(Q)optimal tradeoff between accuracy and computational cost -requires only O(Nmode) ab initio Hessian calculationsfirst readily available numerically stable procedure forgenerating sextic force fields in rectilinear normal modecoordinatesability to accurately model anharmonic fundamentals forsemi-rigid molecules with up to 15 atomsIntroduction Electronic structure PES construction PES validation OutlookWhere to from here?Transform into localized modes→ lower order transformations, expansions required?Transform kinetic energy operator rather than PES?Provide ability to specify ‘spectator modes’ to be modelledusing second order expansion in curvilinear normal modecoordinatesonly generate higher derivatives along ‘active modes’reduce computational cost to N7basisNactiveImprove stability of interpolated PES: transforming into highorder expansions in Cartesian space avoids singularitiesassociated with expanding PES in asymptotically-divergentinternal coordinates.Introduction Electronic structure PES construction PES validation OutlookWhere to from here?Transform between different internal coordinate systems toexploit asymptotic behaviourConstruct and test hybrid force fields using different levels ofelectronic structure theoryHarder than it sounds to do rigorously and cheaply - complexseries of transformations required to find normal modecoordinates and equilibrium geometry for hybrid force fieldanalytically from component partsCould cheat and assume geometry and normal modecoordinates stay approximately constant and do literalreplacement of force constants

A transformative approach to constructing anharmonic force fields

https://ir.canterbury.ac.nz/bitstream/10092/15213/2/Presentation.pdf

A transformative approach to constructing anharmonic force fields

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