Ab initio molecular dynamics: Basic Theory and Advanced Met
|
|
- Hilary Hill
- 5 years ago
- Views:
Transcription
1 Ab initio molecular dynamics: Basic Theory and Advanced Methods Uni Mainz October 30, 2016
2
3 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.
4 Route 1 Route 2
5 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)
6 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.
7 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.
8 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
9 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.
10 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.
11 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 / ]
12 Route 1 Route 2 I I 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)
13 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.
14 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.
15 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
16 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.
17 Route 1 Route 2
18 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)
19 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.
20 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.
21 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.
22 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)
23 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 }).
24 Route 1 Route 2
25 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.
26 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.
27 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.
28 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
29 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
30
31 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)
32 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
33 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 >.
34 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)
35 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)
36 Energy Conservation Electrons do not heat-up, but fluctuate with same frequency as V e Nuclei drag the electrons E phys is essentially constant
37 Deviation from Born-Oppenheimer surface Deviation of forces in CP dynamics from the true BO forces small and/but oscillating.
38 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
39 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
40 Loss of adiabaticity: the bad cases Due to the presence of the vacancy there is a small gap in the system.
41 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.
42 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
43 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) Q eẋ 2 e + 2E kin,0 x e Q Rẋ 2 R + gk B Tx R (78)
44 : examples
45 : examples
46 : examples
47 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.
48 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
49 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)
50 Car-Parrinello vs Born-Oppenheimer dynamics
51 : energy conservation
52 : timing Timing in CPU seconds and energy conservation in a.u./ps for
53 CP of liquid water: Energy conservation CPMD-800-NVE-64 CPMD-400-NVE-128 CPMD-800-NVT-64 J. Phys. Chem. B 2004, 108,
54 CP of liquid water: structure Oxygen-oxygen radial distribution BOMD-NVE (solid line) CPMD-NVE CPMD-NVE CPMD-NVE 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,
55 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 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!
56 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.
57 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)
58 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
59 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
60 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.
Ab initio molecular dynamics
Ab initio molecular dynamics Molecular dynamics Why? allows realistic simulation of equilibrium and transport properties in Nature ensemble averages can be used for statistical mechanics time evolution
More informationEnergy and Forces in DFT
Energy and Forces in DFT Total Energy as a function of nuclear positions {R} E tot ({R}) = E DF T ({R}) + E II ({R}) (1) where E DF T ({R}) = DFT energy calculated for the ground-state density charge-density
More informationAb initio molecular dynamics. Simone Piccinin CNR-IOM DEMOCRITOS Trieste, Italy. Bangalore, 04 September 2014
Ab initio molecular dynamics Simone Piccinin CNR-IOM DEMOCRITOS Trieste, Italy Bangalore, 04 September 2014 What is MD? 1) Liquid 4) Dye/TiO2/electrolyte 2) Liquids 3) Solvated protein 5) Solid to liquid
More informationAb initio Molecular Dynamics Born Oppenheimer and beyond
Ab initio Molecular Dynamics Born Oppenheimer and beyond Reminder, reliability of MD MD trajectories are chaotic (exponential divergence with respect to initial conditions), BUT... With a good integrator
More informationDensity Functional Theory: from theory to Applications
Density Functional Theory: from theory to Applications Uni Mainz May 14, 2012 All electrons vs pseudopotentials Classes of Basis-set Condensed phase: Bloch s th and PBC Hamann-Schlüter-Chiang pseudopotentials
More informationAb initio molecular dynamics and nuclear quantum effects
Ab initio molecular dynamics and nuclear quantum effects Luca M. Ghiringhelli Fritz Haber Institute Hands on workshop density functional theory and beyond: First principles simulations of molecules and
More informationAb initio molecular dynamics : BO
School on First Principles Simulations, JNCASR, 2010 Ab initio molecular dynamics : BO Vardha Srinivasan IISER Bhopal Why ab initio MD Free of parametrization and only fundamental constants required. Bond
More informationAb Initio Molecular Dynamics: Theory and Implementation
John von Neumann nstitute for Computing Ab nitio Molecular Dynamics: Theory and mplementation Dominik Marx and Jürg Hutter published in Modern Methods and Algorithms of Quantum Chemistry, Proceedings,
More informationMolecular Dynamics. Park City June 2005 Tully
Molecular Dynamics John Lance Natasa Vinod Xiaosong Dufie Priya Sharani Hongzhi Group: August, 2004 Prelude: Classical Mechanics Newton s equations: F = ma = mq = p Force is the gradient of the potential:
More informationAccelerated Quantum Molecular Dynamics
Accelerated Quantum Molecular Dynamics Enrique Martinez, Christian Negre, Marc J. Cawkwell, Danny Perez, Arthur F. Voter and Anders M. N. Niklasson Outline Quantum MD Current approaches Challenges Extended
More informationMD Thermodynamics. Lecture 12 3/26/18. Harvard SEAS AP 275 Atomistic Modeling of Materials Boris Kozinsky
MD Thermodynamics Lecture 1 3/6/18 1 Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order
More informationIV. Classical Molecular Dynamics
IV. Classical Molecular Dynamics Basic Assumptions: 1. Born-Oppenheimer Approximation 2. Classical mechanical nuclear motion Unavoidable Additional Approximations: 1. Approximate potential energy surface
More informationTime reversible Born Oppenheimer molecular dynamics
Time reversible Born Oppenheimer molecular dynamics Jianfeng Lu Mathematics Department Department of Physics Duke University jianfeng@math.duke.edu KI-Net Conference, CSCAMM, University of Maryland, May
More informationConical Intersections. Spiridoula Matsika
Conical Intersections Spiridoula Matsika The Born-Oppenheimer approximation Energy TS Nuclear coordinate R ν The study of chemical systems is based on the separation of nuclear and electronic motion The
More informationDensity Functional Theory: from theory to Applications
Density Functional Theory: from theory to Applications Uni Mainz May 27, 2012 Large barrier-activated processes time-dependent bias potential extended Lagrangian formalism Basic idea: during the MD dynamics
More informationAb Ini'o Molecular Dynamics (MD) Simula?ons
Ab Ini'o Molecular Dynamics (MD) Simula?ons Rick Remsing ICMS, CCDM, Temple University, Philadelphia, PA What are Molecular Dynamics (MD) Simulations? Technique to compute statistical and transport properties
More informationDiatomic Molecules. 7th May Hydrogen Molecule: Born-Oppenheimer Approximation
Diatomic Molecules 7th May 2009 1 Hydrogen Molecule: Born-Oppenheimer Approximation In this discussion, we consider the formulation of the Schrodinger equation for diatomic molecules; this can be extended
More information23 The Born-Oppenheimer approximation, the Many Electron Hamiltonian and the molecular Schrödinger Equation M I
23 The Born-Oppenheimer approximation, the Many Electron Hamiltonian and the molecular Schrödinger Equation 1. Now we will write down the Hamiltonian for a molecular system comprising N nuclei and n electrons.
More information2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More informationDensity Functional Theory
Density Functional Theory Iain Bethune EPCC ibethune@epcc.ed.ac.uk Overview Background Classical Atomistic Simulation Essential Quantum Mechanics DFT: Approximations and Theory DFT: Implementation using
More informationBorn-Oppenheimer Approximation
Born-Oppenheimer Approximation Adiabatic Assumption: Nuclei move so much more slowly than electron that the electrons that the electrons are assumed to be obtained if the nuclear kinetic energy is ignored,
More informationAn Introduction to Ab Initio Molecular Dynamics Simulations
John von Neumann Institute for Computing An Introduction to Ab Initio Molecular Dynamics Simulations Dominik Marx published in Computational Nanoscience: Do It Yourself!, J. Grotendorst, S. Blügel, D.
More informationMulti-Scale Modeling from First Principles
m mm Multi-Scale Modeling from First Principles μm nm m mm μm nm space space Predictive modeling and simulations must address all time and Continuum Equations, densityfunctional space scales Rate Equations
More informationWhat is Classical Molecular Dynamics?
What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential functions Newton s equations of motion are integrated
More informationProject: Vibration of Diatomic Molecules
Project: Vibration of Diatomic Molecules Objective: Find the vibration energies and the corresponding vibrational wavefunctions of diatomic molecules H 2 and I 2 using the Morse potential. equired Numerical
More informationQuantum Molecular Dynamics Basics
Quantum Molecular Dynamics Basics Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Depts. of Computer Science, Physics & Astronomy, Chemical Engineering & Materials Science, and Biological
More informationModeling of Nanostructures and Materials Jacek A. Majewski Summer Semester 2013 Lecture 6 Lecture Modeling of Nanostructures Molecular Dynamics
Chair of Condensed Matter Physics nstitute of Theoretical Physics Faculty of Physics, Universityof Warsaw Summer Semester 013 Lecture Modeling of Nanostructures and Materials Jacek A. Majewski E-mail:
More informationAb Initio Molecular Dynamic
Ab Initio Molecular Dynamic Paul Fleurat-Lessard AtoSim Master / RFCT Ab Initio Molecular Dynamic p.1/67 Summary I. Introduction II. Classical Molecular Dynamic III. Ab Initio Molecular Dynamic (AIMD)
More informationMolecular energy levels
Molecular energy levels Hierarchy of motions and energies in molecules The different types of motion in a molecule (electronic, vibrational, rotational,: : :) take place on different time scales and are
More informationFirst-principles Molecular Dynamics Simulations
First-principles Molecular Dynamics Simulations François Gygi University of California, Davis fgygi@ucdavis.edu http://eslab.ucdavis.edu http://www.quantum-simulation.org MICCoM Computational School, Jul
More informationNext generation extended Lagrangian first principles molecular dynamics
Next generation extended Lagrangian first principles molecular dynamics Anders M. N. Niklasson Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Dated: June 1, 017) LA-UR-17-3007
More informationAb Initio Molecular Dynamic
Ab Initio Molecular Dynamic Paul Fleurat-Lessard AtoSim Master Ab Initio Molecular Dynamic p.1/67 Summary I. Introduction II. Classical Molecular Dynamic III. Ab Initio Molecular Dynamic (AIMD) 1 - Which
More informationREVIEW. Hamilton s principle. based on FW-18. Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws!
Hamilton s principle Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws! based on FW-18 REVIEW the particle takes the path that minimizes the integrated difference
More informationExact factorization of the electron-nuclear wave function and the concept of exact forces in MD
Exact factorization of the electron-nuclear wave function and the concept of exact forces in MD E.K.U. Gross Max-Planck Institute for Microstructure Physics Halle (Saale) OUTLINE Thanks Exact factorisation
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Advocacy chit Chemistry 350/450 Final Exam Part I May 4, 005. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle
More informationMixed quantum-classical dynamics. Maurizio Persico. Università di Pisa Dipartimento di Chimica e Chimica Industriale
Mixed quantum-classical dynamics. Maurizio Persico Università di Pisa Dipartimento di Chimica e Chimica Industriale Outline of this talk. The nuclear coordinates as parameters in the time-dependent Schroedinger
More informationWhat s the correct classical force on the nuclei
What s the correct classical force on the nuclei E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale) What s the correct classical force on the nuclei or: How to make the Born-Oppenheimer
More informationModified Ehrenfest formalism: A new approach for large scale ab-initio molecular dynamics
Modified Ehrenfest formalism: A new approach for large scale ab-initio molecular dynamics Xavier Iago Andrade Valencia Dpto. Física de Materiales Universidad del País Vasco/Euskal Herriko Unibertsitatea
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationHandout 10. Applications to Solids
ME346A Introduction to Statistical Mechanics Wei Cai Stanford University Win 2011 Handout 10. Applications to Solids February 23, 2011 Contents 1 Average kinetic and potential energy 2 2 Virial theorem
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Chemistry 3502/4502 Final Exam Part I May 14, 2005 1. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle (e) The
More informationAN ACCELERATED SURFACE-HOPPING METHOD FOR COMPUTATIONAL SEMICLASSICAL MOLECULAR DYNAMICS. Laren K. Mortensen
AN ACCELERATED SURFACE-HOPPING METHOD FOR COMPUTATIONAL SEMICLASSICAL MOLECULAR DYNAMICS by Laren K. Mortensen A senior thesis submitted to the faculty of Brigham Young University in partial fulfillment
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 3 The Born-Oppenheimer approximation C.-K. Skylaris Learning outcomes Separate molecular Hamiltonians to electronic and nuclear parts according to the Born-Oppenheimer
More informationCar-Parrinello Molecular Dynamics
Car-Parrinello Molecular Dynamics Eric J. Bylaska HPCC Group Moral: A man dreams of a miracle and wakes up with loaves of bread Erich Maria Remarque Molecular Dynamics Loop (1) Compute Forces on atoms,
More informationDFT and beyond: Hands-on Tutorial Workshop Tutorial 1: Basics of Electronic Structure Theory
DFT and beyond: Hands-on Tutorial Workshop 2011 Tutorial 1: Basics of Electronic Structure Theory V. Atalla, O. T. Hofmann, S. V. Levchenko Theory Department, Fritz-Haber-Institut der MPG Berlin July 13,
More informationThe Particle-Field Hamiltonian
The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and
More informationGear methods I + 1/18
Gear methods I + 1/18 Predictor-corrector type: knowledge of history is used to predict an approximate solution, which is made more accurate in the following step we do not want (otherwise good) methods
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationCondensed matter physics FKA091
Condensed matter physics FKA091 Ermin Malic Department of Physics Chalmers University of Technology Henrik Johannesson Department of Physics University of Gothenburg Teaching assistants: Roland Jago &
More informationWe start with some important background material in classical and quantum mechanics.
Chapter Basics We start with some important background material in classical and quantum mechanics.. Classical mechanics Lagrangian mechanics Compared to Newtonian mechanics, Lagrangian mechanics has the
More informationDensity Functional Theory. Martin Lüders Daresbury Laboratory
Density Functional Theory Martin Lüders Daresbury Laboratory Ab initio Calculations Hamiltonian: (without external fields, non-relativistic) impossible to solve exactly!! Electrons Nuclei Electron-Nuclei
More informationSelf-consistent Field
Chapter 6 Self-consistent Field A way to solve a system of many electrons is to consider each electron under the electrostatic field generated by all other electrons. The many-body problem is thus reduced
More informationSemi-Classical Theory of Radiative Transitions
Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13 Atomic Structure (recap) Time-dependent
More informationQuantum Theory of Matter
Imperial College London Department of Physics Professor Ortwin Hess o.hess@imperial.ac.uk Quantum Theory of Matter Spring 014 1 Periodic Structures 1.1 Direct and Reciprocal Lattice (a) Show that the reciprocal
More informationTheoretical Photochemistry SoSe 2014
Theoretical Photochemistry SoSe 2014 Lecture 9 Irene Burghardt (burghardt@chemie.uni-frankfurt.de) http://www.theochem.uni-frankfurt.de/teaching/ Theoretical Photochemistry 1 Topics 1. Photophysical Processes
More informationIntroduction to DFTB. Marcus Elstner. July 28, 2006
Introduction to DFTB Marcus Elstner July 28, 2006 I. Non-selfconsistent solution of the KS equations DFT can treat up to 100 atoms in routine applications, sometimes even more and about several ps in MD
More informationFIRST PRINCIPLES MOLECULAR DYNAMICS INVOLVING EXCITED STATES AND NONADIABATIC TRANSITIONS
Journal of Theoretical and Computational Chemistry, Vol., No. 2 (2002) 39 349 c World Scientific Publishing Company FIRST PRINCIPLES MOLECULAR DYNAMICS INVOLVING EXCITED STATES AND NONADIABATIC TRANSITIONS
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 9, February 8, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 9, February 8, 2006 The Harmonic Oscillator Consider a diatomic molecule. Such a molecule
More informationVALENCE Hilary Term 2018
VALENCE Hilary Term 2018 8 Lectures Prof M. Brouard Valence is the theory of the chemical bond Outline plan 1. The Born-Oppenheimer approximation 2. Bonding in H + 2 the LCAO approximation 3. Many electron
More informationSolid State Theory: Band Structure Methods
Solid State Theory: Band Structure Methods Lilia Boeri Wed., 11:15-12:45 HS P3 (PH02112) http://itp.tugraz.at/lv/boeri/ele/ Plan of the Lecture: DFT1+2: Hohenberg-Kohn Theorem and Kohn and Sham equations.
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4 6.5 6.6 6.7 The Schrödinger Wave Equation Expectation Values Infinite Square-Well Potential Finite Square-Well Potential Three-Dimensional Infinite-Potential
More informationCP2K: the gaussian plane wave (GPW) method
CP2K: the gaussian plane wave (GPW) method Basis sets and Kohn-Sham energy calculation R. Vuilleumier Département de chimie Ecole normale supérieure Paris Tutorial CPMD-CP2K CPMD and CP2K CPMD CP2K http://www.cpmd.org
More informationAb initio molecular dynamics
Ab initio molecular dynamics Kari Laasonen, Physical Chemistry, Aalto University, Espoo, Finland (Atte Sillanpää, Jaakko Saukkoriipi, Giorgio Lanzani, University of Oulu) Computational chemistry is a field
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationHow to make the Born-Oppenheimer approximation exact: A fresh look at potential energy surfaces and Berry phases
How to make the Born-Oppenheimer approximation exact: A fresh look at potential energy surfaces and Berry phases E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale) Process of vision
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More informationAlgorithms and Computational Aspects of DFT Calculations
Algorithms and Computational Aspects of DFT Calculations Part I Juan Meza and Chao Yang High Performance Computing Research Lawrence Berkeley National Laboratory IMA Tutorial Mathematical and Computational
More information5.1 Classical Harmonic Oscillator
Chapter 5 Harmonic Oscillator 5.1 Classical Harmonic Oscillator m l o l Hooke s Law give the force exerting on the mass as: f = k(l l o ) where l o is the equilibrium length of the spring and k is the
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationCourse SS18. Organization
0-0 Course SS18 Stefan.Goedecker@unibas.ch Organization Lecture with script: http://comphys.unibas.ch/teaching.htm Simple exercises: Traditional analytic problems and numerical problems (on your own laptop?)
More informationBorn-Oppenheimer Approximation in a singular system Prague. O.T.Turgut Bogazici Univ.
Born-Oppenheimer Approximation in a singular system Prague O.T.Turgut Bogazici Univ. June 5, 206 How can we use Born-Oppenheimer in a singular system (if we can)? What is the usual approach: Let us consider
More informationMolecular Mechanics. I. Quantum mechanical treatment of molecular systems
Molecular Mechanics I. Quantum mechanical treatment of molecular systems The first principle approach for describing the properties of molecules, including proteins, involves quantum mechanics. For example,
More informationStructure of Cement Phases from ab initio Modeling Crystalline C-S-HC
Structure of Cement Phases from ab initio Modeling Crystalline C-S-HC Sergey V. Churakov sergey.churakov@psi.ch Paul Scherrer Institute Switzerland Cement Phase Composition C-S-H H Solid Solution Model
More informationAb-initio molecular dynamics: from the basics up to quantum effects Roberto Car Princeton University
Ab-initio molecular dynamics: from the basics up to quantum effects Roberto Car Princeton University Hands-on Tutorial Workshop on Ab-Initio Molecular Simulations, Fritz- Haber-Institut, Berlin, July 12-21,
More informationJournal of Theoretical Physics
1 Journal of Theoretical Physics Founded and Edited by M. Apostol 53 (2000) ISSN 1453-4428 Ionization potential for metallic clusters L. C. Cune and M. Apostol Department of Theoretical Physics, Institute
More informationNon-adiabatic dynamics with external laser fields
January 18, 21 1 MQCD Mixed Quantum-Classical Dynamics Quantum Dynamics with Trajectories 2 Radiation Field Interaction Hamiltonian 3 NAC NAC in DFT 4 NAC computation Sternheimer Implementation Nearly
More informationIntroduction to density functional perturbation theory for lattice dynamics
Introduction to density functional perturbation theory for lattice dynamics SISSA and DEMOCRITOS Trieste (Italy) Outline 1 Lattice dynamic of a solid: phonons Description of a solid Equations of motion
More informationDensity Functional Theory: from theory to Applications
Density Functional Theory: from theory to Applications Uni Mainz November 29, 2010 The self interaction error and its correction Perdew-Zunger SIC Average-density approximation Weighted density approximation
More informationDensity Functional Theory
Density Functional Theory March 26, 2009 ? DENSITY FUNCTIONAL THEORY is a method to successfully describe the behavior of atomic and molecular systems and is used for instance for: structural prediction
More informationBefore we start: Important setup of your Computer
Before we start: Important setup of your Computer change directory: cd /afs/ictp/public/shared/smr2475./setup-config.sh logout login again 1 st Tutorial: The Basics of DFT Lydia Nemec and Oliver T. Hofmann
More informationDensity Functional Theory for Electrons in Materials
Density Functional Theory for Electrons in Materials Richard M. Martin Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign 1 Density Functional Theory for
More informationQuantum mechanics can be used to calculate any property of a molecule. The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is,
Chapter : Molecules Quantum mechanics can be used to calculate any property of a molecule The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is, E = Ψ H Ψ Ψ Ψ 1) At first this seems like
More informationTime-Dependent Density-Functional Theory
Summer School on First Principles Calculations for Condensed Matter and Nanoscience August 21 September 3, 2005 Santa Barbara, California Time-Dependent Density-Functional Theory X. Gonze, Université Catholique
More informationPhysics 202 Laboratory 5. Linear Algebra 1. Laboratory 5. Physics 202 Laboratory
Physics 202 Laboratory 5 Linear Algebra Laboratory 5 Physics 202 Laboratory We close our whirlwind tour of numerical methods by advertising some elements of (numerical) linear algebra. There are three
More informationHigh Energy D 2 Bond from Feynman s Integral Wave Equation
Applying the Scientific Method to Understanding Anomalous Heat Effects: Opportunities and Challenges High Energy D Bond from Feynman s Integral Wave Equation By: Thomas Barnard Sponsored by: Coolesence
More informationIntro to ab initio methods
Lecture 2 Part A Intro to ab initio methods Recommended reading: Leach, Chapters 2 & 3 for QM methods For more QM methods: Essentials of Computational Chemistry by C.J. Cramer, Wiley (2002) 1 ab initio
More informationChapter 3. The (L)APW+lo Method. 3.1 Choosing A Basis Set
Chapter 3 The (L)APW+lo Method 3.1 Choosing A Basis Set The Kohn-Sham equations (Eq. (2.17)) provide a formulation of how to practically find a solution to the Hohenberg-Kohn functional (Eq. (2.15)). Nevertheless
More informationJ07M.1 - Ball on a Turntable
Part I - Mechanics J07M.1 - Ball on a Turntable J07M.1 - Ball on a Turntable ẑ Ω A spherically symmetric ball of mass m, moment of inertia I about any axis through its center, and radius a, rolls without
More informationDept of Mechanical Engineering MIT Nanoengineering group
1 Dept of Mechanical Engineering MIT Nanoengineering group » To calculate all the properties of a molecule or crystalline system knowing its atomic information: Atomic species Their coordinates The Symmetry
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
More informationCPMD Tutorial Atosim/RFCT 2009/10
These exercices were inspired by the CPMD Tutorial of Axel Kohlmeyer http://www.theochem.ruhruni-bochum.de/ axel.kohlmeyer/cpmd-tutor/index.html and by other tutorials. Here is a summary of what we will
More informationLecture: Scattering theory
Lecture: Scattering theory 30.05.2012 SS2012: Introduction to Nuclear and Particle Physics, Part 2 2 1 Part I: Scattering theory: Classical trajectoriest and cross-sections Quantum Scattering 2 I. Scattering
More informationNon Adiabatic Transitions in a Simple Born Oppenheimer Scattering System
1 Non Adiabatic Transitions in a Simple Born Oppenheimer Scattering System George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics, Mathematical Physics, and Theoretical Chemistry
More informationModels for Time-Dependent Phenomena
Models for Time-Dependent Phenomena I. Phenomena in laser-matter interaction: atoms II. Phenomena in laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein p.1 Outline Phenomena
More informationThe electronic structure of materials 1
Quantum mechanics 2 - Lecture 9 December 18, 2013 1 An overview 2 Literature Contents 1 An overview 2 Literature Electronic ground state Ground state cohesive energy equilibrium crystal structure phase
More informationDENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY
DENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY A TUTORIAL FOR PHYSICAL SCIENTISTS WHO MAY OR MAY NOT HATE EQUATIONS AND PROOFS REFERENCES
More information2m 2 Ze2. , where δ. ) 2 l,n is the quantum defect (of order one but larger
PHYS 402, Atomic and Molecular Physics Spring 2017, final exam, solutions 1. Hydrogenic atom energies: Consider a hydrogenic atom or ion with nuclear charge Z and the usual quantum states φ nlm. (a) (2
More informationTime-dependent density functional theory
Time-dependent density functional theory E.K.U. Gross Max-Planck Institute for Microstructure Physics OUTLINE LECTURE I Phenomena to be described by TDDFT Some generalities on functional theories LECTURE
More informationExample questions for Molecular modelling (Level 4) Dr. Adrian Mulholland
Example questions for Molecular modelling (Level 4) Dr. Adrian Mulholland 1) Question. Two methods which are widely used for the optimization of molecular geometies are the Steepest descents and Newton-Raphson
More informationQuantum Mechanical Simulations
Quantum Mechanical Simulations Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, U.S.A. yan.wang@me.gatech.edu Topics Quantum Monte Carlo Hartree-Fock
More information