Ab initio molecular dynamics: Basic Theory and Advanced Methods Uni Mainz October 30, 2016
Bio-inspired catalyst for hydrogen production Ab-initio MD simulations are used to learn how the active site of an enzyme of hydrogenproducing bacteria could be modified so that it remains stably active in an artificial production system. Courtesy of A. Selloni, Princeton.
Route 1 Route 2
Route 1 Route 2 Starting point: the non-relativistic Schrödinger equation for electron and nuclei. i t Φ({r i}, {R I }; t) = HΦ({r i }, {R I }; t) (1) where H is the standard Hamiltonian: 2 H = 2 I 2M I I i + 1 4πɛ 0 = I = I i<j e 2 r i r j 1 4πɛ 0 2 2m e 2 i (2) I,i e 2 Z I R I r i + 1 4πɛ 0 I <J e 2 Z I Z J R I R J (3) 2 2 I 2 2 i + V n e ({r i }, {R I }) 2M I 2m e i (4) 2 2 I + H e ({r i }, {R I }) 2M I (5)
Route 1 Route 2 Route 1: 1) The electronic problem is solved in the time-independent Schröedinger eq. 2) from here the adiabatic approximation for the nuclei is derived, and as a special case the Born-Oppenheimer dynamics. 3) The classical limit leads then to the classical molecular dynamics for the nuclei.
Route 1 Route 2 Goal: derive the classical molecular dynamics. As intermediate two variant of ab initio molecular dynamics are derived. Solve the electronic part for fixed nuclei H e ({r i }, {R I })Ψ k = E k ({R I })Ψ k ({r i }, {R I }) (6) where Ψ k ({r i }, {R I }) are a set of orthonormal solutions, satisfying: Ψ k({r i }, {R I })Ψ l ({r i }, {R I })dr = δ kl (7) Knowing the adiabatic solutions to the electronic problem, the total wavefunction can be expanded as: Φ({r i }, {R I }, t) = Ψ l ({r i }, {R I })χ l ({R I }, t) (8) l=0 Eq.8 is the ansatz introduced by Born in 1951.
Route 1 Route 2 Inserting (8) into Eq.(1) (multiplying from the left by Ψ k ({r i}, {R I }) and integrating over all the electronic coordinates r) we obtain: [ ] 2 2 I + E k ({R I }) χ k + C kl χ l = i 2M I t χ k (9) I l where: [ C kl = Ψ k ] 2 2 I Ψ l dr + (10) 2M I I + 1 { Ψ M k[ i I ]Ψ l dr}[ i I ] (11) I is the exact non-adiabatic coupling operator. I The adiabatic approximation to the full problem Eq.9 is obtained considering only the diagonal terms: C kk = 2 Ψ 2M k 2 I Ψ k dr (12) I I
Route 1 Route 2 [ I ] 2 2 I + E k ({R I }) + C kk ({R I }) 2M I χ k = i t χ k (13) The motion of the nuclei proceed without changing the quantum state, k, of the electronic subsystem during the evolution. The coupled wavefunction can be simplified as: Φ({r i }, {R I }, t) Ψ k ({r i }, {R I })χ k ({R I }, t) (14) the ultimate simplification consist in neglecting also the correction term C kk ({R I }), so that [ ] 2 2 I + E k ({R I }) χ k = i 2M I t χ k (15) I This is the Born-Oppenheimer approximation.
Route 1 Route 2 On each Born-Oppenheimer surface, the nuclear eigenvalue problem can be solved, which yields a set of levels (rotational and vibrational in the nuclear motion) as illustrated in the figure.
Route 1 Route 2 The next step is to derive the classical molecular dynamics for the nuclei. The route we take is the following: χ k ({R I }, t) = A k ({R I }, t)exp[is k ({R I }, t)/ ] (16) the amplitude A k and the phase S k are both real and A k > 0. Next we substitute the expression for χ k into eq. 15. [ I 2 2M I 2 I + E k ({R I }) ] A k ({R I }, t)exp[is k ({R I }, t)/ ] = (17) i t (A k({r I }, t)exp[is k ({R I }, t)/ ]) I 2 2M I I ( I (A k exp[is k / ])) + E k A k exp[is k / ] = (19) I i A k t exp[is k/ ] + i i S k t A kexp[is k / ] 2 ( ) i I I A k exp[is k / ] + A k 2M I S kexp[is k / ] + E k A k exp[is k / ] = (20) i A k t exp[is k/ ] S k t A kexp[is k / ]
Route 1 Route 2 I I 2 2 2 I A k i I A k 2M I 2M I I S k I I ( ) 2 2 i A k ( S k ) 2 2M I I 2 2M I I A k i I S k (21) 2 2M I A k i 2 S k + E k A k = i A k t Separating the real and the imaginary parts, we obtain: S k t A k S k t + I A k t + I 1 2M I ( S k ) 2 + E k = I 1 M I I A k I S k + I 2 2M I 2 I A k A k (22) 1 2M I A k 2 S k = 0 (23)
Route 1 Route 2 If we consider the equation for the phase: S k t + I 1 2M I ( S k ) 2 + E k = I 2 2M I 2 I A k A k (24) it is possible to take the classical limit 0 which gives the equation: S k t + I 1 2M I ( S k ) 2 + E k = 0 (25) which is isomorphic to Hamilton-Jacobi of classical mechanics: with the classical Hamiltonian function S k t + H k({r I }, { I S k }) = 0 (26) H k ({R I }, {P I }) = T ({P I }) + V k ({R I }) (27) With the connecting transformation: P I = I S k.
Route 1 Route 2 Route 2: 1) Maintain the quantum-mechanical time evolution for the electrons introducing the separation for the electronic and nuclear wf function in a time-dependent way 2) Time-dependent self consistent field (TDSCF) approach is obtained. 3) Ehrenfest dynamics (and as special case Born-Oppenheimer dynamics) 4) The classical limit leads then to the classical molecular dynamics for the nuclei.
Route 1 Route 2 It is possible to follow an alternative route in order to maintain the dynamics of the electron. Φ({r i }, {R I }, t) Ψ({r i }, t)χ({r I }, t)exp[ i t t 0 Ẽ e (t )dt ] (28) where: Ẽ e = Ψ ({r i }, t)χ ({R I }, t)h e Ψ({r i }, t)χ({r I }, t)drdr (29) Inserting this separation ansatz (single determinant ansatz) into i t Φ({r i}, {R I }; t) = HΦ({r i }, {R I }; t) and multiplying from the left by Ψ and by χ we obtain
Route 1 Route 2 +{ i Ψ t = i 2 2m e 2 i Ψ (30) χ ({R I }, t)v ne ({r i }, {R I })χ({r I }, t)dr}ψ (31) +{ i χ t = I 2 2M I 2 i χ (32) Ψ ({r i }, t)h e ({r i }, {R I })Ψ({r i }, t)dr}χ (33) This set of time-dependent Schröedinger equations define the basis of time-dependent self-consistent field (TDSCF) method. Both electrons and nuclei moves quantum-mechanically in time-dependent effective potentials.
Route 1 Route 2
Route 1 Route 2 Using the same trick as before of writing we obtain S k t + I χ k ({R I }, t) = A k ({R I }, t)exp[is k ({R I }, t)/ ] (34) 1 ( S k ) 2 + 2M I A k t + I Ψ H e Ψdr = I 1 M I I A k I S k + I 2 2M I 2 I A k A k (35) 1 2M I A k 2 S k = 0 (36) And in the classical limit 0 S k t + I 1 2M I ( S k ) 2 + Ψ H e Ψdr = 0. (37)
Route 1 Route 2 The Newtonian equation of motion of the classical nuclei are: dp I = I Ψ H e Ψdr (38) dt or M I RI (t) = I V E e ({R I (t)}) (39) here the nuclei move according to classical mechanics in an effective potential V E e ({R I (t)}), often called the Ehrenfest potential.
Route 1 Route 2 The nuclei move according to classical mechanics in an effective potential (Ehrenfest potential) given by the quantum dynamics of the electrons obtained by solving the time-dependent Schröedinger equation for the electrons. +{ i Ψ t = i 2 2m e 2 i Ψ χ ({R I }, t)v ne ({r i }, {R I })χ({r I }, t)dr}ψ (40) Note: the equation (37) still contains the full quantum-mechanics nuclear wavefunction χ({r I }, t). The classical reduction is obtained by: χ ({R I }, t)r I χ({r I }, t)dr R I (t) (41) for 0.
Route 1 Route 2 The classical limits leads to a time-dependent wave equation for the electrons i Ψ t = i 2 2m e 2 i Ψ + V ne ({r i }, {R I })Ψ (42) i Ψ t = H e({r i }, {R I })Ψ({r i }, {R I }) (43) Feedback between the classical and quantum degrees of freedom is incorporated in both direction, even though in a mean field sense. These equations are called Ehrenfest dynamics in honor to Paul Ehrenfest who was the first to address the problem of how Newtonian classical dynamics of point particles can be derived from Schröedinger time-dependent wave equation.
Route 1 Route 2 Difference between Ehrenfest dynamics and Born-Oppenheimer molecular dynamics: In ED the electronic subsystem evolves explicitly in time, according to a time-dependent Schröedinger equation In ED transition between electronic states are possible. This can be showed expressing the electronic wavefunction in a basis of electronic states Ψ({r i }, {R I }, t) = c l (t)ψ l ({r i }, {R I }, t) (44) where l=0 c l (t) 2 = 1 (45) l=0 and one possible choice for the basis functions {Ψ k } is obtained solving the time-independent Schröedinger equation: H e ({r i }, {R I })Ψ k ({r i }, {R I }) = E k ({R I })Ψ k ({r i }, {R I }). (46)
Route 1 Route 2 The Ehrenfest dynamics reduces to the Born-Oppenheimer molecular dynamics if only one term is considered in the sum: Ψ({r i }, {R I }, t) = c l (t)ψ l ({r i }, {R I }, t) l=0 namely: Ψ({r i }, {R I }, t) = Ψ 0 ground state adiabatic wavefunction (47) This should be a good approximation if the energy difference between Ψ 0 and the first excited state Ψ 1 is large everywhere compared to the thermal energy scale K B T. In this approximation the nuclei move on a single adiabatic potential energy surface, E 0 ({R I }).
Route 1 Route 2
Route 1 Route 2 Classical trajectory calculations on global potential energy surfaces In BO one can think to fully decouple the task of generating classical nuclear dynamics from the task of computing the quantum potential energy surface. E 0 is computed for many different {R I } data points fitted to analytical function Newton equation of motion solved on the computed surfaces for different initial conditions Problem: dimensionality bottleneck. It has been used for scattering and chemical reactions of small systems in vacuum, but is not doable when nuclear degrees of freedom increase.
Route 1 Route 2 Force Field - based molecular dynamics One possible solution to the dimensionality bottleneck is the force field based MD. V E e V FF e = N v 1 (R I ) + I =1 N I <J<K N v 2 (R I, R J ) + (48) I <J v 3 (R I, R J, R K ) +... (49) The equation of motion for the nuclei are: M I R I (t) = I V FF e ({R I (t)}). (50) The electrons follow adiabatically the classical nuclear motion and can be integrated out. The nuclei evolve on a single BO potential energy surface, approximated by a few body interactions.
Route 1 Route 2 Ehrenfest molecular dynamics To avoid the dimensionality bottleneck the coupled equations: [ i Ψ t = i M I R I (t) = I H e (51) ] 2 2 i + V ne ({r i }, {R I })χ({r I }, t) Ψ (52) 2m e can be solved simultaneously. The time-dependent Schröedinger equation is solved on the fly as the nuclei are propagated using classical mechanics.
Route 1 Route 2 Using Ψ({r i }, {R I }, t) = c l (t)ψ l ({r i }, {R I }) l=0 the Ehrenfest equations reads: M I R I (t) = I c k (t) 2 E k (53) = k k c k (t) 2 I E k + k,l c k c l(e k E l )d kl I (54) i ċ k (t) = c k (t)e k i I c k (t)d kl (55) where the non-adiabatic coupling elements are given by D kl = Ψ k t Ψ ldr = Ṙ l Ψ k lψ l = Ṙ l d kl. (56) I I
Combine the advantages of Ehrenfest and BO dynamics Integrate the equation of motion on a longer time-step than in Ehrenfest, but at the same time take advantage of the smooth time evolution of the dynamically evolving electronic subsystem
CP dynamics is based on the adiabatic separation between fast electronic (quantum) and slow (classical) nuclear motion. They introduced the following Lagrangian L CP = I 1 2 M I Ṙ2 I + i where Ψ 0 = 1/ N!det{φ i } and µ is the fictious mass of the electrons. µ < φ i φ i > < Ψ 0 H e Ψ 0 > +constraints (57)
the associated Euler-Lagrange equations are: d L = L dt Ṙ I R I (58) d L = L dt φ φ i i (59) from which the Car-Parrinello equations of motion: M I RI (t) = R I < Ψ 0 H e Ψ 0 > + R I {constraints} (60) µ φ i (t) = δ δφ i < Ψ 0 H e Ψ 0 > + δ δφ {constraints} (61) i
For the specific case of the Kohn-Sham Theory the CP Lagrangian is: L CP = I 1 2 M I Ṙ2 I + i µ < φ i φ i > < Ψ 0 H KS e Ψ 0 > + i,j Λ ij (< φ i φ j > δ ij ) (62) and the Car-Parrinello equations of motion: M I RI (t) = R I µ φ i (t) = He KS φ i + j < Ψ 0 H KS e Ψ 0 > (63) Λ ij φ j (64) The nuclei evolve in time with temperature I M I Ṙ2 I ; the electrons have a fictitious temperature i µ < φ i φ i >.
Why does the CP method work? Separate in practice nuclear and ionic motion so that electrons keep cold, remaining close to min {φi } < Ψ 0 H e Ψ 0 >, namely close to the Born-Oppenheimer surface f (ω) = 0 cos(ωt) i < φ i ; t φ i ; 0 > dt (65)
Energy Conservation E cons = I 1 2 M I Ṙ2 I + i µ < φ i φ i > + < Ψ 0 H KS e Ψ 0 > (66) E phys = 1 2 M I Ṙ2 I + < Ψ 0 He KS Ψ 0 >= E cons T e (67) I V e = < Ψ 0 He KS Ψ 0 > (68) T e = i µ < φ i φ i > (69)
Energy Conservation Electrons do not heat-up, but fluctuate with same frequency as V e Nuclei drag the electrons E phys is essentially constant
Deviation from Born-Oppenheimer surface Deviation of forces in CP dynamics from the true BO forces small and/but oscillating.
How to control of adiabaticity? In a simple harmonic analysis of the frequency spectrum yields 2(ɛ i ɛ j ) ω ij = µ (70) where ɛ i and ɛ j are the eigenvalues of the occupied/unoccupied orbitals of the Kohn-Sham Hamiltonian. The lowest possible electronic frequency is: ωe min E gap µ. (71) The highest frequency ω max e E cut µ. (72) Thus maximum possible time step µ t max. (73) E cut
In order to guarantee adiabatic separation between electrons and nuclei we should have large ω min e ω max n ω max n. and E gap depend on the physical system, so the parameter to control adiabaticity is the mass µ. However the mass cannot be reduced arbitrarily otherwise the timestep becomes too small. Alternatively if t is fixed and µ is chosen µ too small: Electrons too light and adiabacity will be lost µ too large: Time step eventually large and electronic degrees of freedom evolve too fast for the Verlet algorithm
Loss of adiabaticity: the bad cases Due to the presence of the vacancy there is a small gap in the system.
Loss of adiabaticity: the bad cases In this system the gap is periodically opened (up to 0.3 ev) and nearly closed at short distances. The electrons gain kinetic energy in phase with the ionic oscillations.
Zero or small electronic gaps: thermostatted electrons One way to (try to) overcome the problem in coupling of electronic and ionic dynamics is to thermostat also the electrons (Blöchl & Parrinello, PRB 1992) Thus electrons cannot heat up; if they try to, thermostat will adsorb the excess heat Target fictitious kinetic energy E kin,0 instead of temperature Mass of thermostat to be selected appropriately: Too light: Adiabacity violated (electrons may heat up) Too heavy: Ions dragged excessively Note: Introducing the thermostat the conserved quantity changes
M I RI (t) = R I µ φ i (t) = He KS φ i + j < Ψ 0 H KS e Ψ 0 > M I Ṙ I ẋ R (74) Λ ij φ j µ φ i ẋ e (75) in blue are the frictious terms governed by the following equations: [ ] Q e ẍ e = 2 µ φ 2 E kin,0 Q R ẍ R = 2 i [ I ] 1 2 M I Ṙ2 1 2 gk BT The masses Q e and Q R determines the time scale for the thermal fluctuations. The conserved quantity is now: E tot = I 1 2 M I Ṙ2 I + i µ < φ i φ i > + < Ψ 0 H KS e Ψ 0 > (76) (77) + 1 2 Q eẋ 2 e + 2E kin,0 x e + 1 2 Q Rẋ 2 R + gk B Tx R (78)
: examples
: examples
: examples
The CP forces necessarily deviate from the BO The primary effect of µ makes the ions heavier No effect on the thermodynamical and structural properties, but affect the dynamical quantities in a systematic way (vibrational spectra) φ i (t) = φ 0 i (t) + δφ i (t) (79) Inserting this expression into the CP equations of motions: F CP I,α(t) = F BO I,α(t) + i µ{< φ i φ0 i > R I,α < φ 0 i R I,α φ i >} + O(δφ 2 i ) (80) the additional force is linear in the mass µ so that it vanishes properly as µ 0.
The new equations of motion: where, in the isolated atom approximation, (M I + µ M I ) R I = F I (81) µ M I = 2 3 µe kin I = 2 m e 3 2 < φ I j 2 2 j φ I j >> 0 (82) 2m e j is an unphysical mass, or drag, due to the fictitious kinetics of the electrons for a system where electrons are strongly localized close to the nuclei there more pronounced renormalization effect
Example: Vibrations in water molecule To correct for finite-µ effects: Perform simulation for different µ-values and extrapolate for µ 0. use mass renormalization according to: ω BO = ω CP 1 + µm M (83)
Car-Parrinello vs Born-Oppenheimer dynamics
: energy conservation
: timing Timing in CPU seconds and energy conservation in a.u./ps for
CP of liquid water: Energy conservation CPMD-800-NVE-64 CPMD-400-NVE-128 CPMD-800-NVT-64 J. Phys. Chem. B 2004, 108, 12990.
CP of liquid water: structure Oxygen-oxygen radial distribution BOMD-NVE (solid line) CPMD-NVE CPMD-NVE-400-64 CPMD-NVE-400-128 CPMD-NVT-400 CPMD-NVT-800 CP2K-MC-NVT The radial distribution functions are correct and independent of the method used. J. Phys. Chem. B 2004, 108, 12990.
Car-Parrinello method: Summary Molecular dynamics can be used to perform real-time dynamics in atomistic systems Verlet algorithm yields stable dynamics (in CPMD implemented algorithm velocity Verlet) Born-Oppenheimer dynamics: Max time step 1 fs (highest ionic frequency 2000-3000 cm 1 ) Car-Parrinello dynamics: Max time step 0.1 fs Car-Parrinello method can yield very stable dynamical trajectories, provided the electrons and ions are adiabatically decoupled The method is best suited for e. g. liquids and large molecules with a wide electronic gap The speed of the method is comparable or faster than using Born-Oppenheimer dynamics and still more accurate (i. e. stable) One has to be careful with the choice of µ!
Forces calculation Efficient calculation of the forces One possibility is the numerical evaluation of : F I = Ψ 0 H e Ψ 0 (84) which can be costly and not so accurate. So analytical derivative is desiderable: I Ψ 0 H e Ψ 0 = (85) Ψ 0 I H e Ψ 0 + I Ψ 0 H e Ψ 0 + Ψ 0 H e I Ψ 0 (86) For the Hellmann Feymann Theorem: F HFT I = Ψ 0 I H e Ψ 0 (87) if Ψ 0 is en exact eigenfunction of H e. in numerical calculation that is not the case and contribution to the forces also arise from variation of the wf with respect to atomic positions.
We need to calculate the additional forces coming form the wf contribution. Let s start with Ψ 0 = 1/N!detφ i (88) where the orbitals φ i can be expanded as: φ i = ν c iν f ν (r, {R I }) (89) Two contributions to the forces emerge from the variation of the wf: I φ i = ν ( I c iν )f ν (r, {R I }) + ν c iν ( I f ν (r, {R I }) (90)
We calculate now the contribution due to the incomplete-basis-set correction. Using: I φ i = ν ( I c iν )f ν (r, {R I }) + ν c iν ( I f ν (r, {R I }) (91) into we obtain: I Ψ 0 H e Ψ 0 + Ψ 0 H e I Ψ 0 (92) ciµc iν ( I f µ H e f ν + f µ H e I f ν ) + (93) iµν ( I ciµh µν c iν + ciµh µν I c iν ) (94) iµν Making use of H µν c iν = ɛ i S µν c iν (95) ν ν
We obtain: ciµc iν ( I f µ H e f ν + f µ H e I f ν )+ ɛ i ( I ciµs µν c iν +ciµs µν I c iν ) iµν iµν (96) which can be rewritten as: ciµc iν ( I f µ H e f ν + f µ H e I f ν ) + ɛ i S µν I (ciµc iν ) (97) iµν iµν and ciµc iν ( I f µ H e f ν + f µ H e I f ν ) + (98) iµν ɛ i [ I S µν ciµc iν I S µν ciµc iν ] (99) iµν i µν And finally: ciµc iν ( I f µ H e ɛ i f ν + f µ H e ɛ i I f ν ) (100) iµν
In the case of plane waves Pulay forces vanish - this is due to the fact that plane wave do not depend on the atomic coordinates - it is only true if number of plane waves is kept fixed. An additional contribution to the forces comes from the non-self-consistency F NSC I = dr( I n)(v SCF V NSC ) (101) Such a force vanishes when the self consistency is reached, meaning Ψ 0 is the exact wavefunction within the subspace spanned by the finite basis set. In Car-Parrinello, as well as in Ehrenfest MD there is no minimization full self-consistency is not required, so this force is irrelevant.