in the previous section. The isobaric–isothermal or NpT ensemble is obtained bycombining barostats <strong>and</strong> thermostats, see Ref. 389 for a general formulation <strong>and</strong>Ref. 391 for reversible integration schemes.An important practical issue in isobaric ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong> simulationsis related to problems caused by using a finite basis set, i.e. “incomplete–basis–set” or Pulay–type contributions to the stress, see also Sect. 2.5. Using afinite plane wave basis (together with a finite number of k–points) in the presenceof a fluctuating cell 245,211 one can either fix the number of plane waves or fix theenergy cutoff; see Eq. (122) for their relation according to a rule–of–thumb. Aconstant number of plane waves implies no Pulay stress but a decreasing precisionof the calculation as the volume of the supercell increases, whence leading to a systematicallybiased (but smooth) equation of state. The constant cutoff procedurehas better convergence properties towards the infinite–basis–set limit 245 . However,it produces in general unphysical discontinuities in the total energy <strong>and</strong> thus in theequation of state at volumes where the number of plane waves changes abruptly,see e.g. Fig. 5 in Ref. 211 .Computationally, the number of plane waves has to be fixed in the frameworkof Car–Parrinello variable–cell <strong>molecular</strong> <strong>dynamics</strong> 94,202,201,55 , whereas the energycutoff can easily be kept constant in Born–Oppenheimer approaches to variable–cell <strong>molecular</strong> <strong>dynamics</strong> 681,682 . Sticking to the Car–Parrinello technique a practicalremedy 202,55 to this problem consists in modifying the electronic kinetic energyterm Eq. (173) in a plane wave expansion Eq. (172) of the Kohn–Sham functionalE KS Eq. (75)E kin = ∑ ∑ 1f i2 |G|2 |c i (q)| 2 , (283)i qwhere the unscaled G <strong>and</strong> scaled q = 2πg reciprocal lattice vectors are interrelatedvia the cell h according to Eq. (111) (thus Gr = qs) <strong>and</strong> the cutoff Eq. (121)is defined as (1/2) |G| 2 ≤ E cut for a fixed number of q–vectors, see Sect. 3.1.The modified kinetic energy at the Γ–point of the Brillouin zone associated to thesupercell reads∑ 1f i ∣ ˜G ( A, σ, Ecut) ∣ eff 2 ∣ |ci (q)| 2 (284)Ẽ kin = ∑ i q∣ ˜G ( A, σ, Ecut) ∣ eff 2 ∣ = |G| 2 + A2{1 + erf[ ]}12 |G|2 − Ecuteffσ(285)where A, σ <strong>and</strong> Ecut eff are positive definite constants <strong>and</strong> the number of scaled vectorsq, that is the number of plane waves, is strictly kept fixed.In the limit of a vanishing smoothing (A → 0; σ → ∞) the constant number ofplane wave result is recovered. In limit of a sharp step function (A → ∞; σ → 0)all plane waves with (1/2) |G| 2 ≫ Ecut eff have a negligible weight in Ẽ kin <strong>and</strong> arethus effectively suppressed. This situation mimics a constant cutoff calculation atan “effective cutoff” of ≈ Ecut eff within a constant number of plane wave scheme. Forthis trick to work note that E cut ≫ Ecut eff has to be satisfied. In the case A > 0 theelectronic stress tensor Π given by Eq. (189) features an additional term (due to99
changes in the “effective basis set” as a result of variations of the supercell), whichis related to the Pulay stress 219,660 .Finally, the strength of the smoothing A > 0 should be kept as modest as possiblesince the modification Eq. (284) of the kinetic energy leads to an increase of thehighest frequency in the electronic power spectrum ∝ A. This implies a decreaseof the permissible <strong>molecular</strong> <strong>dynamics</strong> time step ∆t max according to Eq. (55). Itis found that a suitably tuned set of the additional parameters (A, σ, Ecut) eff leadsto an efficiently converging constant–pressure scheme in conjunction with a fairlysmall number of plane waves 202,55 . Note that the cutoff was kept strictly constantin applications of the Born–Oppenheimer implementation 679 of variable–cell<strong>molecular</strong> <strong>dynamics</strong> 681,682 , but the smoothing scheme presented here could beimplemented in this case as well. An efficient method to correct for the discontinuitiesof static total energy calculations performed at constant cutoff was proposedin Ref. 211 . Evidently, the best way to deal with the incomplete–basis–set problemis to increase the cutoff such that the resulting artifacts become negligible on thephysically relevant energy scale.4.3 Beyond Ground States4.3.1 IntroductionExtending ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong> to a single non–interacting excited state isstraightforward in the framework of wavefunction–based methods such as Hartree–Fock 365,254,191,379,281,284,316,293 , generalized valence bond (GVB) 282,283,228,229,230 ,complete active space SCF (CASSCF) 566,567 , or full configuration interaction(FCI) 372 approaches, see Sect. 2.7. However, these methods are computationallyquite dem<strong>and</strong>ing – given present–day algorithms <strong>and</strong> hardware. Promising stepsin the direction of including several excited states <strong>and</strong> non–adiabatic couplings arealso made 385,386,387,71 .Density functional theory offers an alternative route to approximately solvingelectronic structure problems <strong>and</strong> recent approaches to excited–state propertieswithin this framework look promising. In the following, two limiting <strong>and</strong> thuscertainly idealistic situations will be considered, which are characterized by either• many closely–spaced excited electronic states with a broad thermal distributionof fractional occupation numbers, or by• a single electronic state that is completely decoupled from all other states.The first situation is encountered for metallic systems with collective excitations orfor materials at high temperatures compared to the Fermi temperature. It is notedin passing that associating fractional occupation numbers to one–particle orbitals isalso one route to go beyond a single–determinant ansatz for constructing the chargedensity 458,168 . The second case applies for instance to large–gap <strong>molecular</strong> systemswhich complete a chemical reaction in a single excited state as a result of e.g. avertical homo / lumo or instantaneous one–particle / one–hole photoexcitation.100
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John von Neumann Institute for Comp
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2500Number200015001000CP PRL 1985AI
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The goal of this section is to deri
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¡the Newtonian equation of motion
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Ehrenfest molecular dynamics is cer
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the Car-Parrinello approach 108 , s
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According to the Car-Parrinello equ
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Figure 4. (a) Comparison of the x-c
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Up to this point the entire discuss
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parameters are those used to repres
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in terms of a linear combination of
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structure calculations, see e.g. Re
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“unbound electrons” dissolved i
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Table 1. Timings in cpu seconds and
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stressed that the energy conservati
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see e.g. the discussion following E
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from a set of one-particle spin orb
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is used, which represents exactly a
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2.8.3 Generalized Plane WavesAn ext
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disposable parameters that can be o
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The index i runs over all states an
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f(G) are related by three-dimension
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- Page 128 and 129: AcknowledgmentsOur knowledge on ab
- Page 130 and 131: 57. M. Bernasconi, G. L. Chiarotti,
- Page 132 and 133: Superiore di Studi Avanzati (SISSA)
- Page 134 and 135: 175. E. Ermakova, J. Solca, H. Hube
- Page 136 and 137: 244. H. Goldstein, Klassische Mecha
- Page 138 and 139: 313. T. Ikeda, M. Sprik, K. Terakur
- Page 140 and 141: 384. N. A. Marks, D. R. McKenzie, B
- Page 142 and 143: 442. S. Nosé and M. L. Klein, Mol.
- Page 144 and 145: 502. L. M. Ramaniah, M. Bernasconi,
- Page 146 and 147: 562. F. Shimojo, K. Hoshino, and Y.
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682. R. M. Wentzcovitch, Phys. Rev.