Ab initio thermodynamics of solids under the quasiharmonic ... · MolEng Thermodynamics Analytical EOS DFT EECConclusionsAckBib Content 1 Molecular engineering tools 2 ab initio Thermodynamics

MolEng Thermodynamics Analytical EOS DFT EEC Conclusions Ack Bib

Ab initio thermodynamics of solids under thequasiharmonic approach

Víctor Luaña (†) & Alberto Otero-de-la-Roza (‡)(†) Departamento de Química Física y Analítica, Universidad de Oviedo

(‡) Universidad de California at Merced

VI Escuela de Altas PresionesVI EAP, Oviedo, May 20–24, 2013

V. Luaña & A. Otero-de-la-Roza () Thermodynamics of Solids 6th EAP, Oviedo 2013 1 / 40


Content

1 Molecular engineering tools

2 ab initio Thermodynamics of Crystals

3 Analytical Equations of State

4 DFT calculation of the electronic structure

5 Empirical Energy Corrections

6 Conclusions

7 Acknowledgements

8 Bibliography



1 Molecular engineering toolsThe need for molecular engineeringSome Applications in Geophysics and Material Science





6 Conclusions

7 Acknowledgements

8 Bibliography



The need for a molecular engineering tool

• Columbus, Ohio (May 11, 2013) Chemical Abstracts Ser-vice (CAS) records 64,576,136 substances in CAS REG-ISTRY. • The 50th entry was registered on sept. 7th, 2009.• The 60th entry was registered on may. 24th, 2011.• Some 12000 new entries are added everyday.• The properties of most substances will remain unkonwnunless there is some previous hint of their importance.

• (January 1, 2013) The Cambridge Structural Database(CSD) records the crystal structure of 638598 differentcompounds (mostly organic and organometallic).• (April 2013) The Inorganic Crystal Structure Database(ICSD) contains 161 030 structures.• The structure of most compounds will remain unknown,including the effect of p and T conditions.



“A theory is the more impressive the greater the simplicity of itspremises, the more different kinds of things it relates, and themore extended its area of applicability. Therefore the deep im-pression that classical thermodynamics made upon me. It is theonly physical theory of universal content which I am convincedwill never be overthrown, within the framework of applicability ofits basic concepts.”

Albert Einstein (ca.1949)



Some Applications in Geophysics and Material Science






2 ab initio Thermodynamics of CrystalsThe Standard Model of Material SciencePractical calculation of thermodynamic properties under QHA




6 Conclusions

7 Acknowledgements

8 Bibliography



The Standard Model of Material Science

The thermodynamic properties of a solid under a given pressure (p) and temperature(T) are determined by the general Gibbs energy:

G(p, T, x) = E(V, x) + pV(x) + Avem(V, T, x) (1)

where x represents the internal geometry of the crystal, V is the cell volume, E is theelectronic energy. The Helmholtz energy Avem term includes the contribution fromlattice vibrations, free electrons in metals, magnons in paramagnetic materials, etc.

E(V, x) and p = −dE/dV is the source of the static approach, quite useful for a firstanalysis of the high pressure solid properties.Three steps are fundamental in the full thermal treatment:

Fit an analytical Equation of State (EOS) to the E(V, x) discrete set of data.Objectives: (1) determine the equilibrium properties (V0, E0, B0, B′0, ...); (2)interpolate; (3) derive the curve; and perhaps (4) extrapolate.

Scale the data if agreement with the experiment is critical.

Determine thermal effects (including zero point energy).



Practical calculation of thermodynamic properties under QHA

HA (harmonic approximation): Given an equilibrium configuration of the crystal(i.e. zero forces: ∇qE = 0, where qi =

√Mi(Ri − R0

i ) for atom i) each atom movesin an harmonic potential (K = ∇q ⊗∇qE).

QHA (quasiharmonic approximation): The force constant matrix depends onthe crystal volume: K(V).

At difference from HA, the QHA solid expands when the temperature is raised.

Density Functional Perturbation Theory [1] produces analitically the ab initio Kmatrix.

Frequencies and vibrational modes (phonons) are obtained by diagonalizing K.Sampling the frequencies on the first Brillouin zone produces the phonon-Densityof States (φ-DOS).

Vibrational free energy in QHA:

Avib(V, x, T) =∫ ∞

0g(ω;V, x)

[~ω2

+ kBT ln(

1− e−~ω/kBT)]

dω. (2)



MgO

-147.4

-147.2

-147.0

-146.8

-146.6

-146.4

-146.2

-146.0

40 60 80 100 120 140 160E

(R

y)V (bohr

3)

LDA

PBE

0

100

200

300

400

500

600

700

Γ X Γ L X W L

Fre

quency (

cm

-1)

0.00

0.01

0.01

0.01

0.02

0.03

0.03

0.04

0.04

0.04

0.05

0 100 200 300 400 500 600 700 800

DO

S (

1/c

m-1

)

Frequency (cm-1

)

LDAPBE at V0

PBE at Vexp





3 Analytical Equations of StateThe Murnaghan EOSThe Birch-Murnaghan EOS familyNon-linear fitting and convergence of resultsBM as a polynomial strain familyBM linear fitting procedureAn indefinite number of polynomial strain familiesAverage of polynomials and predicted error barsDo the different average strain families converge and agree?Resampling error bars



6 Conclusions

7 Acknowledgements

8 Bibliography



The Murnaghan EOS

Murnaghan historical paper [2] is based on the principle of conservation of masscombined with Hooke’s law for an inifinitesimal variation of stress in the solid. Anequivalent result, however, can be simply obtained by assumming a linear variation ofthe bulk modulus with pressure, B(p) = B0 + B′0p, to integrate B = −V(∂p/∂V)T :

V(p) = V0

(1 +

B′0B0

p)−1/B′0

(3)

which can be inverted to

p(V) =B0

B′0

[(V0

V

)B′0

− 1

]. (4)

The hydrostatic work equation, dW = −pdV, can then be used to integrate:

E = E0 +B0VB′0

[(V0/V)B′

0

B′0 − 1+ 1

]− B0V0

B′0 − 1(5)

for the volume dependent energy under null temperature conditions.



The Birch-Murnaghan EOS family

Origin: Taylor expansion of the strain energy E =

n∑k=0

ckf k.

Eulerian strain: f =12

[(Vr/V)

2/3 − 1]

Conditions: limf→0

{V;E; p = − dE

dV;B = −V

dpdV

;B′ =dBdp

; ...

}={

V0;E0; 0;B0;B′0; ...}

Results: Vr = V0;c0 = E0;c1 = 0;c2 = 9

2 V0B0;c3 = 9

2 V0B0(B′0 − 4);c4 = 3

8 V0B0{9[B0B′′0 + (B′0)2]− 63B′0 + 143}; ...

BM2: Let x = (V/V0).

E = E0 +98

B0V0(x−2/3 − 1)2, p =

32

B0

(x−7/3 − x−5/3

). (6)

BM3:

E = E0 +9

16V0B0

(x2/3 − 1)2

x7/3{x1/3(B′0 − 4)− x(B′0 − 6)}, ... (7)



Non-linear fitting and convergence of results

Test system: Rock-salt phase of MgO.Calculations: QUANTUM ESPRESSO [3] LDA using plane-waves and norm-conservingpseudopotentials. Very strict conditions: cutoff energy (80 Ry), k-mesh (shifted4× 4× 4 Monkhorst-Pack grid) and q-mesh (6× 6× 6).Data points: linear grid of 129 points in the volume range 72–143 bohr3.

Equilibrium properties of the rock-salt phase of MgO.EOS V0 (bohr3) B0 (GPa) B′0 B′′0 (GPa−1)Murn. 124.885 177.23 3.505Vinet 124.783 165.42 4.524BM3 124.824 170.04 4.116 −0,023 6PT3 124.637 160.80 5.212 −0,090 4BM4 124.821 169.27 4.171 −0,025 4PT4 124.847 169.24 4.089 −0,016 2

The dispersion of fitting parameters is very large (17 GPa for B0). The five parameterEOS (BM4 and PT4) are more in agreement, but we can do better with our excellentinput data. Converging BM3 and BM4 is easy, BM5 is difficult, who wants to try BM10?



BM as a polynomial strain family

The big question: If the BM family comes from assuming a polynomial E(f ) equation,why do not use a linear fit for the polynomial coefficients?

Is the result of linear and nonlinear fittings identical?model fit V0 E0 B0 B′0 B′′0BM3 nonlin. 124.823661 -34.344311 170.0414 4.115715 —BM3 lin. 124.823661 -34.344311 170.041394 4.115715 -0.023630BM4 nonlin. 124.821340 -34.344283 169.2683 4.171190 -0.025412BM4 lin. 124.821340 -34.344283 169.268283 4.171190 -0.025412BM12 lin. 124.824369 -34.344285 169.319792 4.151925 -0.024118

• Linear BM fittings can be done precise and efficiently to any order requiredby the data • Linear fittings can be generalized to many strain definitions.



BM linear fitting procedure

1 Get the eulerian strain f = ((V/Vr)−2/3−1)/2. Vr is a reference volume.

2 Linear fitting to the E(f ) =∑n

0 ckf k polynomial.3 Get minimum of E(f ) to determine the equilibrium properties.4 Get the derivatives Enf ≡ dnE/df n.5 Determine fnV = dnf/dVn. For the eulerian strain:

f(n+1)V = −(3n + 2)fnV/(3V), f1V = −(1/3V)(Vr/V)2/3, n = 1, 2, 3, ... (8)

6 Obtain EnV = dnE/dVn:

E1V = E1f f1V ; (9)

E2V = f 21V E2f + f2V E1f ; (10)

E3V = f 31V E3f + 3f1V f2V E2f + f3V E1f ; (11)

E4V = f 41V E4f + 6f 2

1V f2V E3f + (4f1V f3V + 3f 23V)E2f + f4V E1f ; ... (12)

7 Finally

p = −E1V ; B = VE2V ; B′p = −VE3V

E2V− 1; B′′ = E−3

2V [V(E4V E2V − E23V) + E3V E2V ]; ...

(13)



An indefinite number of polynomial strain families

strain EOS f f(n+1)V f1V s

eulerian BM12

(x−2/3−1

)(3n+2)sfnV − x−2/3

3V− 1

3V

lagrangian Thomson12

(x2/3−1

)(3n−2)sfnV − x2/3

3V− 1

3Vnatural PT

13

ln x nsfnV1

3V− 1

V

infinitesimal Bardeen 1−x−1/3 (3n+1)sfnV(1− f )4

3Vr− 1

3V

x3 x3 (3n−1)sfnVx1/3

3V− 1

3VV V 0 1 0

x = V/Vr, η = x1/3.BM: Birch-Murnaghan [2, 4, 5]; Thomson [6]; PT: Poirier-Tarantola [7]; Bardeen [8].



Average of polynomials and predicted error barsIdea:

Produce a better polynomial by averaging several strain polynomials according to theirefficiency.

The square sum of residuals: Si =∑

k[Ek − E(Vk)]2.

wi = [Ni/ni][Si/Smin], where Ni is the number of data, ni is the degree of thepolynomial, and Smin = mini Si.

A normalized probability distribution is defined as Pi = e−wi/(∑

k e−wk ) (method 1),or Pi = e−w2

i /(∑

k e−w2k ) (method 2).

The Pi are used to: (1) average the polynomial coefficients, thus defining aaverage polynomial; and (2) average the equilibrium properties of the individualpolynomials, determining in this way the probability distribution of the properties(mean, standard deviation, skew, kurtosis, etc).

Use bootstrap sampling when data is noisy.

This method can be used with arbitrary strain polynomial families.



Do the different average strain families converge and agree?

0.82100.82150.82200.82250.82300.82350.82400.82450.82500.8255

V0-

124

(boh

r3 )

0.20.30.40.50.60.70.80.91.01.1

B0-

169

(GP

a)

4.104.124.144.164.184.204.22

B0’

3 4 5 6 7 8 9 10 11 12 13 14 15 16 BM

PT

Lag

Inf

-0.038-0.036-0.034-0.032-0.030-0.028-0.026-0.024-0.022-0.020-0.018

B0’’ (

GP

a-1)

order n

0.82050.82100.82150.82200.82250.82300.82350.82400.82450.8250

V0-

124

(boh

r3 )

0.10.20.20.20.30.30.40.40.50.5

B0-

169

(GP

a)

4.134.144.154.164.174.184.194.204.214.224.234.24

B0’

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-0.034-0.032-0.030-0.028-0.026-0.024-0.022-0.020

B0’’ (

GP

a-1)

BM average up to order n



The results are stable (within the predicted error bars) for the many experimentsthat we have done: (1) decreasing the 129 volume points to 65, 33, 17, orincreasing them to 257, 1025, 2049; (2) moving the grid points; (3) changing thevolume range up to 1/2 or 2 times; ...

The different strain families agree.

For a good set of data, the error bars are small, orders of magnitude better thanthe errors currently assumed by using BM3, BM4 or Vinet.

Equilibrium properties of MgOEOS V0 (bohr3) B0 (GPa) B′0 B′′0 (GPa−1)BM3 124.823661 170.0414 4.115715BM4 124.821340 169.2683 4.171190Vinet 124.783430 165.4184 4.524436avg14BM 124.82372(83) 169.314(20) 4.1593(58) -0.02461(85)

Contrary to the current opinion, the average of strain polynomials produces asignificative value for B0, B′0, and B′′0 . In the case of B′′′0 the error bar is larger thanthe predicted value.



Resampling error bars

The resampling method (aka Monte Carlo statistics) is a standard technique for (a)estimating the precision of sample statistics by using subsets of available data(jackknifing) or drawing randomly with replacement from a set of data points(bootstrapping); and (b) validating models by using random subsets (bootstrapping,cross validation).

MgO: BM6 fitting to a resampling of the 129 data points.sample points V0 (bohr3) B0 (GPa) B′0 B′′0 (GPa−1)

102 50–76 124.82385(36) 169.3117(78) 4.1618(13) -0.02489(17)103 48–79 124.82381(34) 169.3105(82) 4.1620(12) -0.02491(15)104 45–84 124.82381(34) 169.3107(75) 4.1620(11) -0.02491(15)105 39–89 124.82381(34) 169.3107(75) 4.1620(11) -0.02491(15)106 39–91 124.82381(34) 169.3107(75) 4.1620(11) -0.02491(15)

avg14BM 124.82372(83) 169.314(20) 4.1593(58) -0.02461(85)

The error bars of the strain polynomials average method are conservative.






4 DFT calculation of the electronic structureDensity Functional TheoryThe Kohn-Sham methodThe Jacob’s ladder metaphor


6 Conclusions

7 Acknowledgements

8 Bibliography



The solid state physics approach: Density Functional Theory

Much is due to the seminal work of J.C. Slater and coworkers (Xα and APW methods,for instance), Wigner-Seitz (cellular method), Philips-Kleinman (pseudopotentials), . . .However, nowadays is common to start from the Hohenberg-Kohn theorems (1964):

HK1: The ground state electron density, ρ0(r1), of a bound system of interactingelectrons in some external potential, Vext, determines this potential uniquely.In consequence, the ground state electron density is sufficient to construct the fullHamilton operator and hence to calculate, in principle, any ground state propertyof the system. In other words, any ground state property can be expressed interms of the ground state electron density ρ0.Tipically Vext represents the Coulomb potential of a set of nuclei.The theorem can be extended to the time-dependent domain to developtime-dependent density functional theory (TDDFT), which can be used to describeexcited states.

HK2: The functional E for the ground state energy is minimized by the groundstate electron density ρ0. I.e.: E [ρ] ≥ E [ρ0] for every trial electron density ρ.This enables calculating ρ0 variationally.



The Kohn-Sham method

DFT is, in principle, an orbital-free method: minimize the energy

E[ρ] = T[ρ]+∫R3

Vext(r)ρ(r)dr+12

∫R6

ρ(r)ρ(r′)|r− r′| drdr′+Exc[ρ] = T[ρ]+En[ρ]+EH[ρ]+Exc[ρ]

(14)by varying ρ over all densities containing N electrons. However, T[ρ] an essentialcomponent of the total energy (T[ρ] = −E[ρ] by the virial theorem), is unknown, andHK1 demands that the electron density comes from a well behaved wave function(N-representability).Most DFT calculations follow the Kohn-Sham route: ρ(r) =

∑λ nocc

λ |ψλ|2, where thespinorbitals are the result of solving, self-consistently, the KS equations:{

−12∇2

r + Vn(r) + VH(r, [ρ]) + Vxc[ρ]

}ψλ(r) = ελψλ(r) (15)

Vn: nuclear electrostatic potential (or sum of pseudopotentials of the chemicalkernels);

VH(r) =∫R3

ρ(r′)dr′

|r− r′| : Coulomb potential due to the (possibly valence) electron

density.Vxc: Exchange and correlation term.



There is no systematic way of improving the xc functional...

... but J. P. Perdew has popularized the Jacob’s ladder metaphor.

Jacob’s ladder by Shalomde Safed (1887-1980)

• First rung (LDA/LSDA) [Local (spin) density approach]:Based on the Quantum Monte Carlo solution of an homogeneouselectron gas. Ex: VWN91.

• Second rung (GGA) [Generalized gradient approach]: De-termined by the value of the electron density and its gradient oneach point. Ex: PW91, PBE, ...

• Third rung (meta-GGA): Depends on the local density, gradi-ent, and kinetic energy. Ex: TPSS, ...

• Fourth rung, Hybrid functionals: Mix of several kinds. Typi-cally, the Hartree-Fock exchange (exact exchange) is mixed withGGA. In other words, the functional depends on the occupied or-bitals. Ex: B3LYP, ...

• Fifth rung, Double hybrid functionals: Depends on the vir-tual and occupied orbitals. Ex: XYG3, ...

Increasing cost: L(S)DA ≈ GGA < meta-GGA <<? hybrid<<< double-hybrid.







5 Empirical Energy CorrectionsSystematic errors in the DFT resultsEmpirical correction of the systematic errorsDoes the EEC really works?Any thermodynamic property can be obtainedMgO as a high pT pressure scalePerformance of the method for other crystalsAvailability

6 Conclusions

7 Acknowledgements

8 BibliographyV. Luaña & A. Otero-de-la-Roza () Thermodynamics of Solids 6th EAP, Oviedo 2013 27 / 40


DFT calculations on MgO

Problems that prevent true accuracy:Choice of xc: source of uncertainty in DFT.Known trends in solids: LDA overbinds, PBE underbinds.No systematic ways of improving the results. Wait for a hypothetical perfectfunctional?Fitting of the E(V) and A(V; T) curves. Solved!!!.

100

105

110

115

120

125

130

135

0 5 10 15 20 25 30 35 40 45 50

V (

bohr3

)

p (GPa)

LDAPBE

Speziale (2001)Li (2006)

Tange (2009)

120

125

130

135

140

145

150

0 500 1000 1500 2000

T (K)

Fiquet (1999)Sinogeikin (2000)

Dubrovinsky (1997)



Empirical Energy Corrections (EEC)

The systematic deviations in the exchange-correlation functionals can be corrected byusing one or two accurate experimental results: ambient conditions volume (V0

exp) andbulk modulus (B0

exp).

Kunc and Syassen’s observation: p/B0 vs. V/V0 is transferable, V0 and B0 being thestatic equilibrium volume and bulk modulus, respectively.

E′sta(V)

Bexp=

E′sta

(V V0

Vexp

)B0

(16)

The bpscal EEC is:

Esta(V) = Esta(V0) +BexpVexp

B0V0

[Esta

(V

V0

Vexp

)− Esta(V0)

](17)

The experimental Vexp and Bexp extrapolated to static conditions are missing, and aretreated as parameters to fit the experimental data.



Does the EEC really works?

100

105

110

115

120

125

130

135

0 5 10 15 20 25 30 35 40 45 50

V (

bohr3

)

p (GPa)

LDAPBE

Speziale (2001)Li (2006)

Tange (2009)

120

125

130

135

140

145

150

0 500 1000 1500 2000

T (K)

Fiquet (1999)Sinogeikin (2000)

Dubrovinsky (1997)



100

150

200

250

300

350

400

0 5 10 15 20 25 30 35 40 45 50

Bs (

GP

a)

p (GPa)

80

90

100

110

120

130

140

150

160

170

0 500 1000 1500 2000

T (K)

Anderson (1990)Sinogeikin (2000)



Any thermodynamic property can be obtained (MgO)

The three steps: (1) energy fitting, (2) full QHA, and (3) empirical energy correctionsallow us to predict thermodynamic properties with unprecedented accuracy.

Room (0 GPa and 298.15 K) (0 GPa and 1000 K)PBEcorr expt. PBEcorr expt.

V (bohr3) 126.024(19) 126.025 129.708(84) 129.6Fvib (kJ/mol) 11.4236(47) -31.613(59)S (J/mol K) 27.0028(88) 27.18 82.853(56) 82.24pth (GPa) 2.624(15) 0.717 6.943(67) 4.96BT (GPa) 161.43(50) 161.3 141.2(20) 141BS (GPa) 163.72(50) 163.9 151.6(20) 151.1α (1 · 10−5 K) 3.1002(97) 3.12 4.680(68) 4.47Cv (J/mol K) 36.6821(50) 36.9 48.5530(28) 47.61Cp (J/mol K) 37.2023(57) 37.409 52.132(55) 50.87B′T 4.173(43) 4.53(12)B′′T (GPa−1) -0.0333(28) -0.089(24)γ 1.53435(24) 1.54 1.5751(11) 1.54



MgO as a high pT pressure scale

Magnesium oxide has been proposed as pressure scale for diffraction experiments atextreme pressure and temperature. This requires a very precise knowledge of thep(V, T) EOS.

0

50

100

150

200

80 85 90 95 100 105 110 115 120 125 130

p (

GP

a)

V (bohr3)

3000 K, Tmo=3125 K

This studyTange(2009), VinetTange(2009), BM3

Wu(2008)Speziale(2001)



Performance of the method for other crystals

68 69 70 71 72 73 74 75 76 77 78

0 5 10 15 20 25 30 35 40 45 50

V (

bohr3

)

Occelli (2003)Dewaele (2008)

75.0

75.5

76.0

76.5

77.0

77.5

78.0

78.5

79.0

79.5

0 500 1000 1500 2000

C

Reeber (1996)

75

80

85

90

95

100

105

110

115

0 5 10 15 20 25 30 35 40 45 50

V (

bohr3

)

p (GPa)

Dewaele (2004)Chijioke (2005)

Akahama (2006)

106

108

110

112

114

116

118

120

0 100 200 300 400 500 600 700 800 900

T (K)

Al

Langelaan (1999)



Availability

Any method able to provide static E(V) data and phonon DOS as a function ofvolume can be used. Among the ab initio open source electronic structure codes:Quantum Espresso, abinit, siesta, ...The codes for fitting and for doing the complete analysis described here are opensource and publicly available:• GIBBS2: A new version of the quasi-harmonic model code. I Robust treatment of the staticdata. Comput. Phys. Commun. 182 (2011) 1708. doi:10.1016/j.cpc.2011.04.016.• GIBBS2: A new version of the quasi-harmonic model code. II. Thermal models, featuresand implementation. Comput. Phys. Commun. 182 (2011), 2232–2248,doi:10.1016/j.cpc.2011.05.009.

The technique and results have been described in:• Equations of State in Solids. Fitting theoretical data, possibly including noise and jumps.Comput. Theor. Chem. 975 (2011) 111. doi:10.1016/j.comptc.2011.03.050.• Treatment of first-principles data for predictive quasiharmonic thermodynamics of solids:The case of MgO. Phys. Rev. B 84 (2011) 024109.doi:10.1103/PhysRevB.84.024109.• Equations of state and thermodynamics of solids using empirical corrections in thequasiharmonic approximation. Phys. Rev. B 84(2011) 184103,doi:10.1103/PhysRevB.84.184103.


doi:10.1016/j.cpc.2011.04.016

doi:10.1016/j.cpc.2011.05.009

doi:10.1016/j.comptc.2011.03.050

doi:10.1103/PhysRevB.84.024109

doi:10.1103/PhysRevB.84.184103


Conclusions

The conventional procedure of fitting nonlinearly an analytical low order EOS(BM3, Vinet, ...) introduces a significant error in the equilibrium properties and inthe derivatives calculated from E(V) theoretical data.

The average of strain polynomial method: is efficient, numerically stable, can beextended to any order required by the data, and provides an estimation of theaccuracy, in the form of error bars, that can be translated to any thermodynamicproperty.

An empirical energy correction (EEC) can be used to remove the systematic errorin DFT calculations due, mainly, to the defects in the exchange-correlationfunctional.

The quasiharmonic approximation, improved with the EEC and a careful fit of theE(V) and A(V) curves, can predict the thermodynamic properties of a crystal in avery wide range of pressures and temperatures.

Genetic algorithms and other methods of random sampling can be used to predictnew metastable phases and conformations of solids.

Chemical bonding properties in a solid do depend heavily on the pressure and, toa lesser extent, the temperature conditions.



Acknowledgements

Funds have been provided by the Spanish Ministerio de Ciencia e Innovaciónunder grant CTQ2009-08376.AOR is indebted to the Spanish Ministerio de Educación for a FPU grant.The research is part of the MALTA Consolider Ingenio initiative (grant no.CSD2007-00045). See http://www.malta-consolider.com/.Original artwork has been provided by Andrea Victoria Luaña Lafuente (aka Anvi).


http://www.malta-consolider.com/


Muchas gracias por su atención ...



... ¿Alguna pregunta?



References I

S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Phonons and related crystal properties fromdensity-functional perturbation theory, Rev. Modern Physics 73 (2001) 515–562.

F. D. Murnaghan, The compressibility of media under extreme pressures, Proc. Natl. Acad. Sci. USA 30(1944) 244–247.

P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti,M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann,C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini,A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov,P. Umari, R. M. Wentzcovitch, QUANTUM ESPRESSO: a modular and open-source software project forquantum simulations of materials, J. Phys.-Condens. Matter 21 (2009) 395502.

F. Birch, Finite elastic strain of cubic crystals, Phys. Rev. 71 (1947) 809–824.

F. Birch, Finite strain isotherm and velocities for single-crystal and polycrystalline NaCl at high pressuresand 300 K, J. Geophys. Res. 83 (1978) 1257–1268.

L. Thomson, On the fourth order anharmonic equation of state of solids, J. Phys. Chem. Solids 31 (1970)2003–2016.

J.-P. Poirier, A. Tarantola, A logarithmic equation of state, Phys. Earth Planet. Int. 109 (1998) 1–8.

J. Bardeen, Compressibilities of the alkali metals, J. Chem. Phys. 6 (1938) 372–378.


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