Chapter 20 Density-Functional Theory of Quantum Freezing and the Helium Isotopes Steven W. Rick , John D. McCoy , and A. D. J . Haymet Downloaded by UNIV OF GUELPH LIBRARY on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch020
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Structural Biochemistry Program, Frederick Biomedical Supercomputing Center, National Cancer Institute-Frederick Cancer Research and Development Center, Frederick, MD 21702 Department of Materials Engineering, New Mexico Institute of Mining and Technology, Socorro, NM 87801 School of Chemistry, University of Sydney, Sydney, New South Wales 2006, Australia 2
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Over the lastfifteenyears density functional theory has lead to new understanding of many properties of the freezing transition. The essence of the theory is a functional Taylor series expansion of the free energy of the solid about that of a reference liquid. In the simplest version of the theory, the reference state is taken to be the coexisting liquid. The series is typically truncated after second order in the density difference between the two phases. In the classical atomic— and original—version of the theory, all the information specific to the system is contained in the second order expansion coefficient, which is related to pair correlation functions of the liquid (1, 2). The theory demonstrated that liquids freeze when they are strongly correlated, in agreement with empirical freezing rules observed in computer simulations (3, 4). An early, successful application of density functional freezing theory was to the Lennard-Jonesfluid(5). The phase diagram for many of the rare gas elements, when plotted in reduced variables obtained by scaling with the Lennard-Jones parameters which contain the length and energy scales of the particular element, are remarkably similar. For example, the triple points of neon, argon, krypton, and xenon a l l lie i n the same region of phase space. Furthermore, the phase dia gram for the Lennard-Jones system, calculated both from simulation and density functional theory, agrees well w i t h the rare gas data. T h e exception is h e l i u m . In terms of its scaled variables, helium freezes at a much lower density than the more classical rare gases (6). A d d i t i o n a l l y , the less massive and therefore more q u a n t u m mechanical isotope, H e , freezes at a lower density than H e . Q u a n t u m effects are therefore promoting freezing and stablizing the solid phase. O n the other h a n d , quantum effects decrease correlations between particles. T h i s last fact is problematic for density functional freezing theory, which gets a l l its infor m a t i o n about the system from liquid-state correlation functions. T h e influence of q u a n t u m mechanics, if only interparticle correlations are considered, w i l l be to increase the freezing density. 3
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0097-6156/96/0629-0286$15.00/0 © 1996 American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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DFT of Quantum Freezing
Therefore, from these two facts—quantum effects decrease the freezing den sity and also decrease interparticle correlations—we know a priori that classical atomic freezing theory w i l l fail to predict the proper trend among H e , H e and the classical l i m i t . Recognizing that the free energy Taylor series is an expansion of the excess free energy, defined as the difference between the system of interest and an ideal system at the same thermodynamic conditions, improvements can be made to the theory by choosing a better ideal system. Classically, it suffices to choose a set of non-interacting classical particles. For the quantum case, then, it is sensible to choose a set of non-interacting q u a n t u m particles. T h i s choice should make the series more rapidly convergent, hopefully by the second order t e r m , which is where we typically truncate. B y using the F e y n m a n path integral representation of quantum mechanics, w i t h its well-known isomorphism between the quantum particle and a ring polymer, the quantum freezing theory has many features i n common w i t h the density functional theory as applied to actual polymers (7).
Downloaded by UNIV OF GUELPH LIBRARY on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch020
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Quantum Density Functional Theory of Freezing T h i s derivation of the quantum freezing functional follows closely that found i n Reference 8, which the interested reader should consult for further details. T h e first step i n the derivation of the free energy is a functional Taylor series expansion of the excess H e l m h o l t z free energy about the l i q u i d state i n powers of the singlet density difference,
where the subscripts S and L refer to the solid and l i q u i d states, pi is the bulk l i q u i d density, p(r) is the solid singlet density Δ ^ ( Γ ) = p(r) — PL and A is the free energy of the ideal system. T h e natural variables of the H e l m h o l t z ensemble are the temperature, T , the volume, V , and p. T h e ideal system has the same values of T , V , and p as the interacting system; this means that for the ideal system of the solid phase, which has a spatially varying density, an external field must be placed on the ideal system to produce the same density. In order to study phase coexistence, it is more convenient to use the G r a n d ensemble, which is related to the H e l m h o l t z potential through the Legendre trans form, 0
Ω = Α-|(ΙΓΦ(Γ)/)(Γ),
(2.2)
where Ω is the G r a n d potential, Φ(Γ) = μ — t / ( r ) , μ is the chemical potential, and U ( r ) is an external field. T h e variables p and Φ are conjugate and the following relations h o l d ,
In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
288
CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY
and
< · 2
4 )
U s i n g Equations (2.2) and (2.3) and defining
where β = kT, leads to the following functional for the Grand potential free energy difference, Downloaded by UNIV OF GUELPH LIBRARY on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch020
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ΔΩ>(p)]
il'-ill
= =
A°-A°
+ J dr [ 9
L
v
-
drdv' c(|r -
i AT jj
-
L
v
f
»(P)]P(P) -
y
dr Ψ ° Δ / > ( Γ )
Γ'|)Δ (Γ)Δ ,(Γ') . Ρ
(2.6)
/
Since the natural variables of Ω are Τ , V , and Φ ( Γ ) , the free energy should be a functional of these variables only. However, rather than eliminate p(r) i n favor of these variables, i t is more convenient to consider Ω as a functional of T , V , Φ ( Γ ) , and p(r), w i t h an additional condition which fixes the value of p(r). T h i s condition is that p(r) is given by the value which minimizes Ω * for fixed T , V , and Φ ( Γ ) . T h e asterisk on the Δ Ω * functional denotes that p(r) is a free variable, only when the functional is m i n i m i z e d does it equal the the grand potential difference, ΔΩ.
It is now necessary to choose the ideal system. For classical atomic liquids, the ideal system is commonly chosen to be classical non-interacting particles. For this ideal system, c(r) is the Ornstein-Zernike direct correlation function, />(r)=exp(/?$(r)) and βΑ° = jT A r p ( p ) p n ( p ( p ) - 1 ] .
(2.7)
For a q u a n t u m mechanical ideal system, using the Feynman p a t h integral rep resentation, the singlet density of the ideal non-interacting system is related to Φ ° ( Γ ) through
(
p ν 3(P-l)/2 , , ζ) Α · · / * » · · · dr e «VP)i*°(r) ... *o(rr)] P
J
A ) 1
x
e
+
+
JV
-(irP/A )[(r-r )»+...+(r -rH 2
2
P
?
( .8) 2
where λ = (2nrnkT/h )~ / , m is the mass of the particle, h is P l a n c k ' s con stant, and Ρ is the number of discretizations of the path integral (9, 10). F r o m Equations (2.2) and (2.4), 2
1
2
βA
0
= J dr y
p ( r ) [ * ° ( r ) - 1] .
In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
(2.9)
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For the l i q u i d phase Φ is a constant and therefore /0£,=βχρ(/?Φ£) just as i n he classical case. Substituting E q u a t i o n (2.9) into E q u a t i o n (2.6) leads to /?ΔΩ* = V
+
PL
fiJ dr v
[*
L
- Φ(Γ) - \np + *°(p) - l]^(r) L
" \ IIv ' ^ Downloaded by UNIV OF GUELPH LIBRARY on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch020
kT
drdT C
V
" Ί)ΜΓ)ΔΡ(Γ') Γ
.
(2.10)
A s already noted, / ? Δ Ω * is only the free energy difference when m i n i m i z e d w i t h respect to /?(r), but now we have the added function, Φ ° ( Γ ) , which cannot be eliminated i n favor of p(r) because E q u a t i o n (2.8) cannot be inverted to give Φ°(Γ) as a functional of p(r). Therefore, the / ? Δ Ω * from E q u a t i o n (2.10) is a functional of both p(r) and Φ°(Γ). T h e functional so defined is the actual G r a n d potential free energy difference when | ^ dp(r)
= 0
(111)
= 0.
(2.12)
and
