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Anomalous Transport Mechanisms of Hydronium and Hydroxide Ions in Aqueous Solution Probed by Ab Initio Molecular Dynamics

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Anomalous Transport Mechanisms of Hydronium and Hydroxide Ions in Aqueous Solution Probed by Ab Initio Molecular Dynamics Ab initio molecular dynamics via the Car-Parrinello method: Basic ideas, theory and algorithms Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences New York University, 100 Washington Sq. East New York, NY 10003 1808: “We are perhaps not far removed from the time when we shall be able to submit the bulk of chemical phenomena to calculation.” Joseph Louis Gay-Lussac (1778-1850) “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact solution of these laws leads to equations much to complicated to be soluble.” Paul Dirac on Quantum Mechanics (1929). “Every attempt to refer chemical questions to mathematical doctrines must be considered, now and always, profoundly irrational, as being contrary to the nature of the phenomena.” August Comte, 1830 Motivation • Car-Parrinello is a method for performing molecular dynamics with forces obtained from electronic structure calculations performed “on the fly” as the simulation proceeds. This is known as ab initio molecular dynamics (AIMD). • As a result, AIMD calculations are considerably more expensive than force-field calculations, which only involve evaluation of simple functions of position. • Force fields, although useful, are, with notable exceptions, unable to treat chemical bond breaking and forming events. • Force fields often lack transferability to thermodynamic situations in which they are not designed to work. • Polarization and manybody interactions included implicitly. From ISI Citation Report R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985) Total Cites = 4,812 The “Universal” Hamiltonian M Electrons N Nuclei ˆ ˆ ˆ ˆ ˆ ˆ HTTVVVe n ee en nn Operator Definitions: Electronic: Nuclear: MN ˆˆ122 1 1 TTe i n I 22iI11MI MN1 ZZ VVˆˆ IJ ee nn ˆˆ i jrrˆˆij I J RRIJ Coupling: MN Z Vˆ I en ˆ iI11rRˆiI Molecular energy levels Electron coordinates Nuclear coordinates Notation: r r11,..., rMN R R,..., R x r1,ssz ,1 ,..., r M , z , M 10 szi, , , or , , 01 Complete energy level spectrum: HEˆ (,)(,)x R x R TTVVVEˆ ˆ ()(,)()(,)(,)rˆˆ r R ˆ R ˆ x R x R e n ee en nn Born-Oppenheimer Approximation à la W. H. Flygare, Molecular Structure and Dynamics Mass disparity: MmH 2000 e Quasi adiabatic separability ansatz for wave function: (,)(,)()x R x R R Schrödinger equation separates if II()(,)R x R . Electrons in fixed back- TVVˆ ˆ()(,)(,)()(,)rˆˆ ˆ r R x R R x R ground nuclear geometry R e ee en Nuclei on each electronic TVEˆ()()()()RRRR ˆ ˆ ˆ hypersurface n nn Born-Oppenheimer (electronic) surfaces and nuclear energy levels n ()R ε2 ε1 (no bound levels) Vibrations Rotations ε0 R Classical nuclear motion on an electronic surface Consider the ground-state electronic surface 0 ()R Nuclear Hamiltonian: ˆ ˆ ˆ ˆ H TVn 0 ()()RR nn “Demote” to a classical Hamiltonian: N 2 PI H (,)()()PRRR 0 Vnn I 1 2MI Nuclear motion now given by Hamilton’s equations: HH RPII PRII Classical nuclei (R,P) Quantum electrons 0 (,)xR Hellman-Feynman Theorem Ground-state electronic surface as expectation value: ˆ (e) ˆˆ(e) 0()()()()RRRR 0H 0 HTVV()()(,)Re ee rˆˆ en r R ˆ (e) 0H 0ˆˆ(e) (e) 0 0()()()()()()RRRRRR 0 HH 0 0 RRRRIIII ˆ (e) 0H 0 0 0()RRRRR 0 () 0() 0 () 0 () RRRR IIII 0 Because 0(RRRR ) 0 ( ) 1 0( ) 0 ( ) 0 RI Kohn-Sham density functional theory Except for very small systems, we cannot solve for the exact 01(xx ,...,M ) Density functional theory represents a compromise between accuracy and computational cost. Wave function ansatz: 1()()xx 1 1 M 01(xx ,...,M ) MMM()()xx1 Single-particle orbitals: i(x ) i j ij Electron density: 2 M /2 n()r M d r d r (,...,) x x () x 2 2M 0 1 M i s1 sMz i1 s /2 Kohn-Sham density functional theory Total energy functional: E[{},]RR Ts [{}] E H [] n E xc [] n E ext [,] n Energy definitions: 1 1nn (rr ) ( ) T[{ }] 2 E[ n ] drr d s i i H 2i 2rr ' n()r E[ n ,Rr ] Z d ext I I rR I Ground-state energy via constrained minimization 0 (RER ) min [{ }, ] ij i j ij {} ij, Kohn-Sham equations (i are eigenvalues of ij ) 1 2 VVEEE(r ) ( r ) ( r ) ( r ) 2KS i i i KSn (r ) H xc ext The Born-Oppenheimer Algorithm Electrons Nuclei ,n Start with nuclei Add electrons Compute i i ,n F Add electrons Propagate nuclei a short time Δt with F e.g. Verlet: t2 RRRFIIII(tt ) 2 (0) ( ) (0) MI The Car-Parrinello scheme Avoid explicit minimization with a fictitious adiabatic dynamics for electronic orbitals: Lagrangian (note μ not a mass! It has units of energy x time2): M 1 2 LME i i IRR I , ij i j ij i1,2 I i j Equations of motion: EE i ij j M IR I iIj R Conditions: 1) “Near” Born-Oppenheimer MN1 2 i i M IR I 0 iI112 “Seed” the CP equations of motion with initially minimized orbitals. Energy Conservation in Born-Oppenheimer and Car-Parrinello dynamics CP 5 a.u CP 10 a.u. BO 10-6, 10 a.u. BO 10-6, 100 a.u. CP 10 a.u. BO 10-5, 100 a.u. BO 10-6, 100 a.u. CP 10 a.u. BO 10-4, 100 a.u. System: 8 Silicon atoms Marx and Hutter, Modern Methods and Algorithms of Quantum Chemistry (NIC Series) 1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000) Energy conservation timing comparison Marx and Hutter, Modern Methods and Algorithms of Quantum Chemistry (NIC Series) 1, J. Grotendorst, ed. (Forschungszentrum, Jülich, 2000) System: 8 Silicon atoms Adiabatic Dynamics Consider a simple 2 degree-of-freedom system: p pR R mm R p F ( R , ) Fbath ( p , bath, ) pRRRR F( R , ) F bath ( p , bath, ) bath, G( bath, , p , T ) bath,RRR G( bath, , p , TR ) Adiabatic conditions: m mRR T T R Analysis of the dynamics Full phase-space vector (,,,,,)p pRR R bath, bath, evolves according to iL Liouville operator: p pR iL FR(,)(,)()() FRRRR iLTiLTbath, bath, m p mRR R p Subdivision of Liouville operator: p iLref , iL bath, () T m pR iLref ,RRR iL bath, () T mRR iL iLref , F (,)(,) R FR R pp R iL iL iLref,R Analysis of dynamics (cont’d) Evolution of phase space over a time Δt characteristic of nuclear motion: ( te ) iL t (0) Trotter factorization: eiL t eiL t/2 eiLref,R t e iL t /2 O t3 Exact Trotter theorem: n t t t t t FFFF iL t/2RR iLref , 4 n p e lim e4n pRRR e 4 n p e2n e e 4 n p n Evolution of momentum: t /2 p( t / 2) p (0) dt F( R (0), ( t )) RRR0 Analysis of dynamics (cont’d) Time-average equated to phase-space average: ( 1/kBRBR T 1/ k T ) d F( R , ) eVR(,) 21t /2 R dt FR ( R , ( t )) Z ( , ) tR0 VR( ,) de Partition function for slow variable: 2 R / Q( , ) dp dR eRRRpm/2 Z( R , ) RR Adiabatic method for free-energy profiles: [L. Rosso, et al. JCP 116, 4389 (2002)] 11 ARZRPR() ln(,) ln() R Annealing property: T0, 2 pR Q(RRR ; ) dp dR exp min V ( R , ) 2mR TTR 10 mmR 5 Model Problem: 2 1 V(,) R D R2 a 2 2 R 0 2 TTR 10 vRR(0) v ( t ) mmR 300 v(0) v ( t ) Methods: Plane-wave basis sets (periodic box, FFTs) 1iik r k g r 2n 1 2 iikg(r )e c, e g | g | E cut V g L 2 11igr 2 n(r ) n ( g ) e | g | 4 Ecut V g 2 orbitals density EE Car-Parrinello kk ci,,gg k* ij c i M IR I ciI,g j R N l = 0 Eliminating ˆ ZI Vext ()r core electrons I 1 rR I l = 1 M ˆ Eext[] n i V ext i l = 2 i1 n()r Zdr I I rR Nl I ˆ Vpseud vlI rR lm lm I10 l m l Nl vl r R I vll r R I v r R I lm lm I10 l m l N N l1 l vll r RI v l r R I v r R I lm lm I1 I 1 l 0 m l MN E[,{}] n Vˆ dr n() r v r R E [{},] R pseud i pseud i l I NL iI11 Why a real-space basis? • Plane-waves are elegant but scale as N 2M • Slow convergence of plane waves to the basis set limit. • Ease of localizing orbitals. • Ease of representing position-dependent operators. • Exact representation of 2 • Common choice – Gaussians 22 i |rRI | /2 i()(;)(;)r C G r R I G r R I N x y z e ,,, I Selecting a real-space basis (why not Gaussians?) • Retain simplicity of plane waves. • Systematic convergence to the basis-set limit. • Spatially localized for possible linear-scaling. • Position independence and orthonormality. • No BSSE • For flexibility of use, seek noncompact support. • Choice: Discrete variable representations (DVRs). J. C. Light, et al. J. Chem. Phys. 82, 1400 (1985); Edwards, Tuckerman, Friesner, Sorensen, J. Comp. Phys. 110, 82 (1994).R. A. Friesner, Chem. Phys. Lett. 116, 39 (1985); Bacic and Light, Ann. Rev. Phys. Chem. 40, 469 (1989); J. T. Muckerman, Chem. Phys. Lett. 173, 200 (1990); Colbert and Miller, J. Chem. Phys. 96, 1982 (1992); Light and Carrington, Adv.
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