Ab initio molecular dynamics: Theory and Implementation

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John von Neumann Institute for Computing

  In the case of using classical mechanics to describe thedynamics – the focus of the present overview – the limiting step for large systems is 3N−6 plicity 10 discretization points per coordinate implies that of the order of 10 electronic structure calculations are needed in order to map such a global potentialenergy surface. The most extensive discussion is related to the features of the basic Car–Parrinello approach Concerning the depth, the focus of the present discussion is clearly the im- plementation of both the basic Car–Parrinello and Born–Oppenheimer molecular 142 dynamics schemes in the CPMD package .

2 I in Eq. (6) in the limit

  In this case, the coefficients {|c (t)| } (withP 2 kk |c (t)| ≡ 1) describe explicitly the time evolution of the populations (occupa- ⋆ c l6=k tions) of the different states {k} whereas interferences are included via the {c k } k contributions. One possible choice for the basis functions {Ψ } is the adiabatic basis obtained from solving the time–independent electronic Schr¨ odinger equation= E (20) e i I k k I k i I H ({r }; {R })Ψ ({R })Ψ ({r }; {R }) , I where {R } are the instantaneous nuclear positions at time t according to Eq.

1 B

  As a result of this derivation, the essential assumptions underlying classical molecular dynamics become transparent: the electrons follow adiabatically the clas-sical nuclear motion and can be integrated out so that the nuclei evolve on a singleBorn–Oppenheimer potential energy surface (typically but not necessarily given by the electronic ground state), which is in general approximated in terms of few–bodyinteractions. In addition to the derivative of the Hamiltonianitself I e I e ∇ hΨ |H |Ψ i = hΨ |∇ H |Ψ i I Ψ e e I Ψ (63) I there are in general also contributions from variations of the wavefunction ∼ ∇In general means here that these contributions vanish exactly HFT F(64) = − hΨ |∇ I H e |Ψ i I if the wavefunction is an exact eigenfunction (or stationary state wavefunction) of the particular Hamiltonian under consideration.

NSC SCF NSC

F n) V (68) I I = − dr (∇ − V and is governed by the difference between the self–consistent (“exact”) potential or SCFNSC field V and its non–self–consistent (or approximate) counterpart V associ- NSC ; n(r) is the charge density. In summary, the total force needed in ab ated to H einitio molecular dynamics simulations HFT

IBS NSC

  F = F + F + F (69) I I I I comprises in general three qualitatively different terms; see the tutorial article 180 Ref. valence states and the effect of pseudopo- tentials.

NSC SCF

  Thus, the particular implementation underlying the comparison between Car–Parrinello and Born–Oppenheimer molec-ular dynamics is an approximate one from the outset concerning the Car–Parrinello 281,282 part; it can be argued that this was justified in the early papers where the basic feasibility of both the Hartree Fock– and generalized valence bond–based Car– 285 Parrinello molecular dynamics techniques was demonstrated . The second term comes from the fixed externalpotential I I J X X Z Z Z V + ext(78) (r) = − I I J |R − r| |R − R | I I<J in which the electrons move, which comprises the Coulomb interactions between electrons and nuclei and in the definition used here also the internuclear Coulombinteractions; this term changes in the first place if core electrons are replaced by pseudopotentials, see Sect.

2 KS

i ij j − ∇ 2 j X KSH φ (r) = Λ φ (r) , (83) i ij j ej which are one–electron equations involving an effective one–particle Hamiltonian

KS KS KS

  In conventional static density functional or “band structure” calculations this set of equations has to besolved self–consistently in order to yield the density, the orbitals and the Kohn– 487 Sham potential for the electronic ground state . The functional deriva- tive of the Kohn–Sham functional with respect to the orbitals, the Kohn–Shamforce acting on the orbitals, can be expressed as KS δE KS = f H φ , (87) i i e⋆ δφ i which makes clear the connection to Car–Parrinello molecular dynamics, seeEq.

GGA GGA

  x c LDA In the simplest case it is the exchange and correlation energy density ε (n) of an xc interacting but homogeneous electron gas at the density given by the “local” density n(r) at space–point r in the inhomogeneous system. (2) e energy HZ X 1 HF ⋆ 2 E dr ψ (r) + V (r) ψ (r) i ext i [{ψ }] = i − ∇ 2 i Z Z X 1 1 ′ ⋆ ⋆ ′ ′ i j ′ 2|r − r | ij Z Z X 1 1 ′ ⋆ ⋆ ′ ′ j i i j′ 2|r − r | ij yields the lowest energy and the “best” wavefunction within a one–determinant ansatz; the external Coulomb potential V was already defined in Eq.

2 HF

  2.8.3 Generalized Plane Waves 263,264 An extremely appealing and elegant generalization of the plane wave concept consists in defining them in curved ξ–space GPW 1/2 f (ξ) = N det J exp [iG r(ξ)] (101) G i ∂r det J = , j ∂ξ where det J is the Jacobian of the transformation from Cartesian to curvilinear√ 1 2 3 , ξ , ξ ) and N = 1/ Ω as for regular plane coordinates r → ξ(r) with ξ = (ξwaves. Thus, a uni- form grid in curved Riemannian space is non–uniform or distorted when viewed inflat Euclidean space (where g = δ ) such that the density of grid points (or the ij ij “local” cutoff energy of the expansion in terms of G–vectors) is highest in regions 275 close to the nuclei and lowest in vacuum regions, see Fig.

3 Plane waves build a complete and orthonormal basis with the above periodicity

  Schemes on how to choose the k30,123,435 integration points efficiently are available in the literature where also an 179 overview on the use of k–points in the calculation of the electronic structure of solids can be found. 3.2 Electrostatic Energy 3.2.1 General ConceptsThe electrostatic energy of a system of nuclear charges Z at positions R and I I an electronic charge distribution n(r) consists of three parts: the Hartree energy of the electrons, the interaction energy of the electrons with the nuclei and theinternuclear interactions Z Z ′ 1 n(r)n(r ) ′ E ES = dr dr ′ 2|r − r | Z X X 1 Z I Z J I + dr V (r)n(r) + .

12 The Ewald method (see e.g. Ref. ) can be used to avoid singularities in the

  (143) c (r) = − − 3 cc R(R ) I I It is convenient at this point to make use of the arbitrariness in the definition of the core potential and define it to be the potential of the Gaussian charge distributionof Eq. To remove the singularity of the Green’s function at x = 0, G(x) is modified for small x and the error is corrected by using the identity 1 x 1 x c c where r is chosen such, that the short-ranged part can be accurately described by c a plane wave expansion with the density cutoff.

2 Dim. periodic (G /4π)V (G)

  This b has to be compared to the 15N b N log N operations needed for the other Fourier transforms of the charge density and the application of the local potential and the 2 4N N operations for the orthogonalization step. The correction energy can be calculated from in out out ⋆ X n(G) n (G) ∆E (n (G)) tot = −2π Ω − 2 2 G G G 6=0 X in out out ⋆ V (G) (n (G)) , (180)−Ω xc (G) − V xc G in outin out where n and n are the input and output charge densities and V and V the xc xc corresponding exchange and correlation potentials.

2 Z

  To achieve this, E is calculated not from the valence density n(R) xc alone, but from a modified density˜ n(R) = n(R) + ˜ n (R) , (200) core where ˜ n (R) denotes a density that is equal to the core density of the atomic core reference state in the region of overlap with the valence density˜ n (r) = n (r) if r > r ; (201) core core with the vanishing valence density inside r . (176), where E xc is replace by E = E (n + ˜ n ) , (202) xc xc core xc xc core The sum of all modified core densities X I n ˜ (G) = n ˜ (G)S (G) (204) core core I I depends on the nuclear positions, leading to a new contribution to the forces X∂E xc ⋆ I iG V (G)˜ n (G)S (G) , (205) s I = −Ω xc core∂R I,s G and to the stress tensor I X X∂E ∂˜ n (G) xc ⋆ core = V (G) S (G) .

1 F(ξ ) − F(ξ

  ′ the conditioned average in the constraintHamiltonian of the system and h· · · i ξ 589 ensemble . By way of the blue moon ensemble, the statistical average is replaced by a time average over a constrained trajectory with the reaction coordinate fixed ′ at special values, ξ(R) = ξ , and ˙ξ(R, ˙ R) = 0.

2 Z M M ∂R ∂R ∂R ∂R

  N number of atoms at N number of projectors p N number of electronic bands or states b N number of plane-waves PW N number of plane-waves for densities and potentials D N , N , N number of grid points in x, y, and z direction x y z N = N N N total number of grid points x y z In Table 3 the relative size of this variables are given for two systems. p N number of atoms on processor p at p N number of projectors on processor p p p N number of electronic bands or states on processor p b p N number of plane-waves on processor p PW p N number of plane-waves for densities and potentials on processor p D p N , N y , N z number of grid points in x, y, and z direction on processor p x p p N =N N y N z total number of grid points on processor p x The real space grid is only distributed over the x coordinates.

30 Percentage

20 10

2 Number of Processors

  A powerful extensionconsists in also allowing for changes of the shape of the supercell to occur as a result 459,460,461,678 of applying external pressure , including the possibility of non–isotropic 460 external stress ; the additional fictitious degrees of freedom in the Parrinello– 459,460,461 Rahman approach are the lattice vectors of the supercell, whereas the 678 strain tensor is the dynamical variable in the Wentzcovitch approach . 3.1.cut is defined as (1/2) |G| ≤ EThe modified kinetic energy at the Γ–point of the Brillouin zone associated to the supercell reads 2 X X 1 eff 2 ˜ ˜E = f G A, σ, E (284) kin i i cut |c (q)|q 2 i ( " #) 1 2 eff 2 cuteff 2 |G| − E 2 ˜G A, σ, E + A 1 + erf (285) cut = |G| σ eff where A, σ and E are positive definite constants and the number of scaled vectors cut q, that is the number of plane waves, is strictly kept fixed.

2 H s

  Whence, this term is the extension to finite temperatures of the “band–structure energy” (or of the “sum 604,418 of orbital energies” in the analogues Hartree–Fock case ) contribution to the total electronic energy, see Eq. The corresponding one– particle density at the Γ–point is given by X 2 n(r) = f (296) i i (β) |φ (r)| i −1 f i (β) = (1 + exp [β (ǫ i , (297)− µ)]) i where the fractional occupation numbers {f } are obtained from the Fermi–Dirac i distribution at temperature T in terms of the Kohn–Sham eigenvalues {ǫ }.

1 For large–gap systems with well separated electronic states it might be desirable

  1 However, suitable Clebsch–Gordon projections of the mixed states |m i and 2 3 1 state |m i yield another triplet state |t i and the desired first excited singlet or S 214 for the total energy of the S state is given by 1 1 |s i. Four possible determinants |t 1 i, |t 2 i, |m 1 i and |m 2 i as a result of the promotion of a single electron from the homo to the lumo of a closed shell system, see text for further details.

1 N

  The crucial difference compared to the self–consistentapproaches presented above is that the creation of the thermal ensemble and the Several attempts to treat also the electrons in the path integral formulation – instead of using wavefunctions as in the ab initio path integral family – were 606,119,488,449,450 published . Classic solid–state applica-tion of this technique focus on the properties of crystals, such as those of CuCl where anharmonicity and off–center displacements of the Cu along the (111) di-rections were found to be important to describe the crystal structure as a func- 64 647 tion of temperature and pressure .

2 Specific to ab initio molecular dynamics is its capability to describe also

  The ab initio calculations of surface phonons in semiconductor sur- faces can be based on the frozen–phonon, linear–response or nowadays molecular 218 dynamics approaches, see Ref. A review on the structure and energetics of oxide surfaces including molecular processes occur- 235 256 ring on such surfaces is provided in Ref.

2 O

  3 , the diffusion of a single Ga adatom on the GaAs(100)–c(4×4) sur- 367 face , homoepitaxial crystal growth on Si(001) and the low–temperature dynam- 595,611 358,359 , dissociation of an H 2 O molecule on MgO , disso- ics of Si(111)–(7×7) 380691 ciation of Cl 2 on GaAs(110) , chlorine adsorption and reactions on Si(100) , 358 molecular motion of NH 3 on MgO , dynamics and reactions of hydrated α– 289 alumina surfaces , molecular vs. Some applications out of 146,147 this emerging field are the cationic polymerization of 1,2,5–trioxane , the 353,354 initial steps of the dissociation of HCl in water , the formation of sulfuric 421 acid by letting SO react in liquid water or the acid–catalyzed addition of water 3 422 to formaldehyde .

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