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Molecular DFT

Introduction to Molecular Approaches of Density Functional Theory

Jan K. Labanowski (jkl@ccl.net)


Ohio Supercomputer Center, 1224 Kinnear Rd, Columbus, OH 43221-1153

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Hohenberg and Kohn theorems

The field of rigorous density functional theory was born in 1964 with the publication of the Hohenberg and Kohn paper (1964). They proved the following:

I.
Every observable of a stationary quantum mechanical system (including energy), can be calculated, in principle exactly, from the ground-state density alone, i.e., every observable can be written as a functional of the ground-state density.
II.
The ground state density can be calculated, in principle exactly, using the variational method involving only density,

The original theorems refer to the time independent (stationary) ground state, but are being extended to excited states and time-dependent potentials (for review see, e.g., Gross & Kurth, 1994).

How these theorems were derived? By quite an original logic. Within a Born-Oppenheimer approximation, the ground state of the system of electrons is a result of positions of nuclei. This is obvious, if we look at the tex2html_wrap_inline1855 hamiltonian in equation (20). In this hamiltonian, the kinetic energy of electrons ( tex2html_wrap_inline1847 ) and the electron-electron interaction ( tex2html_wrap_inline1853 ) ``adjust'' themselves to the external (i.e., coming from nuclei) potential tex2html_wrap_inline1851 . Once the tex2html_wrap_inline1851 is in place, everything else is, including electron density, which simply adjusts itself to give the lowest possible total energy of the system. The external potential tex2html_wrap_inline1851 is the only variable term in this equation, and everything else depends indirectly on it.

Hohenberg and Kohn posed a more interesting question, which is quite opposite to the traditional logic: Is tex2html_wrap_inline1851 uniquely determined from the knowledge of electron density tex2html_wrap_inline1669 ? Can we find out (in principle, though it need not be easy) where and what the nuclei are, if we know the density tex2html_wrap_inline1669 in the ground state? Is there a precise mapping from tex2html_wrap_inline1669 to tex2html_wrap_inline1851 ? The answer to this question is: Yes!. Actually the mapping is accurate within a constant, which would not change anything, since Schrödinger equations with tex2html_wrap_inline1855 and tex2html_wrap_inline2137 yields exactly the same eigenfunctions, i.e., states (it is easy to prove based on the linear property of the hamiltonian), and the energies will be simply elevated by the value of this const. Note that all energies are known only within some constant, which establishes the frame of reference (e.g., we do not include electron-Mars gravitational attraction within most calculations).

Why was this question so important? Because, if this is true, the knowledge of density provides total information about the system, and formally if we know the density, we know everything there is to known. Since tex2html_wrap_inline1669 determines number of electrons, N:

equation924

and tex2html_wrap_inline2145 determines the tex2html_wrap_inline1851 , the knowledge of total density is as good, as knowledge of tex2html_wrap_inline1695 , i.e., the wave function describing the state of the system. They proved it through a contradiction:

1.
Assume that we have an exact ground state density tex2html_wrap_inline1669 ,
2.
Assume that the ground state is nondegenerate (i.e., there is only one wave function tex2html_wrap_inline1695 for this ground state (though HK theorems can be easily extended for degenerate ground states, see, e.g., Dreizler & Gross, 1990; Kohn, 1985),
3.
Assume that for the density tex2html_wrap_inline1669 there are two possible external potentials: tex2html_wrap_inline1851 and tex2html_wrap_inline2159 , which obviously produce two different hamiltonians: tex2html_wrap_inline1855 and tex2html_wrap_inline2163 , respectively. They obviously produce two different wave functions for the ground state: tex2html_wrap_inline1695 and tex2html_wrap_inline2167 , respectively. They correspond to energies: tex2html_wrap_inline2169 and tex2html_wrap_inline2171 , respectively.
4.
Now, let us calculate the expectation value of energy for the tex2html_wrap_inline2167 with the hamiltonian tex2html_wrap_inline1683 and use variational theorem:

  equation942

5.
Now let us calculate the expectation value of energy for the tex2html_wrap_inline1695 with the hamiltonian tex2html_wrap_inline2179 and use varational theorem:

  equation954

6.
By adding equations (61) and (62) by sides we obtain:

equation967

and it leads to a contradiction.

Since now, we know that tex2html_wrap_inline1669 determines N and tex2html_wrap_inline1851 , it also determines all properties of the ground state, including the kinetic energy of electrons tex2html_wrap_inline2187 and energy of interaction among electrons tex2html_wrap_inline2189 , i.e., the total ground state energy is a functional of density with the following componentsgif:

equation978

Additionally, HK grouped together all functionals which are secondary (i.e., which are responses) to the tex2html_wrap_inline2201 :

  equation983

The tex2html_wrap_inline2203 functional operates only on density and is universal, i.e., its form does not depend on the particular system under consideration (note that N-representable densities integrate to N, and the information about the number of electrons can be easily obtained from the density itself).

The second HK theorem provides variational principle in electron density representation tex2html_wrap_inline1669 gif. For a trial density tex2html_wrap_inline2209 such that tex2html_wrap_inline2211 and for which tex2html_wrap_inline2213 ,

equation1004

where tex2html_wrap_inline2215 is the energy functional. In other words, if some density represents the correct number of electrons N, the total energy calculated from this density cannot be lower than the true energy of the ground state.

As to the necessary conditions for this theorem, there is still some controversy concerning the, so called, representability of density. The N-representability, i.e., the fact that the trial tex2html_wrap_inline2221 has to sum up to N electrons is easy to achieve by simple rescaling. It is automatically insured if tex2html_wrap_inline1669 can be mapped to some wave function (for further discussion see: Parr & Yang, 1989; Gilbert, 1975; Lieb, 1982; and Harriman, 1980). Assuring that the trial density has also tex2html_wrap_inline2193 -representability (usually denoted in the literature as v-representability) is not that easy. Levy (1982) and Lieb (1983) have shown, that there are some ``reasonable'' trial densities, which are not the ground state densities for any possible tex2html_wrap_inline2193 potential, i.e., they do not map to any external potential. Such densities do not correspond therefore to any ground state, and their optimization will not lead to a ground state. Moreover, during energy minimization, we may take a wrong turn, and get stuck into some non v-representable density and never be able to converged to a physically relevant ground state density. For an interesting discussion, see Hohenberg et al. (1990). Assuming that we restrict ourselved only to trial densities which are both N and v representable, the variational principle for density is easly proven, since each trial density tex2html_wrap_inline2221 defines a hamiltonian tex2html_wrap_inline2241 . From the hamiltonian we can derive the corresponding wave function tex2html_wrap_inline2243 for the ground state represented by this hamiltonian. And according to the traditional variational principle, this wave function tex2html_wrap_inline2243 will not be a ground state for the hamiltonian of the real system tex2html_wrap_inline1855 :

equation1020

where tex2html_wrap_inline2249 is the true ground state density of the real system.

The condition of minimum for the energy functional: tex2html_wrap_inline2251 needs to be constrained by the N-representability of density which is optimizedgif. The Lagrange's method of undetermined multipliers is a very convenient approach for the constrained minimization problems. In this method we represent constraints in such a way that their value is exactly zero when they are satisfied. In our case, the N representability constraint can be represented as:

equation1034

These constraints are then multiplied by an undetermined constants and added to a minimized function or functional.

equation1038

where tex2html_wrap_inline2257 is yet undetermined Lagrange multiplier. Now, we look for the minimum of this expression by requiring that its differential is equal to zero (a necessary condition of minimum).

equation1043

Solving this differential equation will provide us with a prescription of finding a minimum which satisfies the constraint. In our case it leads to:

equation1048

since tex2html_wrap_inline2257 and N are constants. Using the definition of the differential of the functional (see, e.g., Parr & Yang, 1989):

equation1053

and the fact that differential and integral signs may be interchanged, we obtain

equation1057

Since integration runs over the same variable and has the same limits, we can write both expressions under the same integral:

equation1065

which provides the condition for constrained minimisation and defines the value of the Lagrange multiplier at minium. It is also expressed here via external potential from equation (65):

  equation1072

Density functional theory gives a firm definition of the chemical potential tex2html_wrap_inline2257 , and leads to several important general conclusions. For review, please refer to Parr & Yang (1989), chapters 4 and 5.

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