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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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Wave Functions


Since inception of quantum mechanics by Heisenberg, Born, and Jordan in 1925, and Schrödinger in 1926, there were basically two competing approaches to find the energy of a system of electrons. One was rooted in statisticial mechanics and the fundamental variable was the total electron density tex2html_wrap_inline1669 , i.e., the number of electrons per unit volume at a given point in space (e.g., in cartesian coordinates: tex2html_wrap_inline1671 ). In this approach, electrons were treated as particles forming a special gas, called electron gas. The special case, the uniform electron gas, corresponds to the tex2html_wrap_inline1673 .

Another approach was to derive the many particle wave function tex2html_wrap_inline1675 (where the tex2html_wrap_inline1677 denotes the coordinates of the 1st electron, tex2html_wrap_inline1679 the 2nd electron, and so on, and t is time) and solve the stationary (time-independent) Schrödinger equation for the system:

  equation47

(where tex2html_wrap_inline1683 is the hamiltonian, i.e., the operator of the total energy for the system), and calculate the set of possible wave functions (called eigenfunctions) tex2html_wrap_inline1685 and corresponding energies (eigenvalues) tex2html_wrap_inline1687 . The eigenfunctions have to be physically acceptable, and for finite systems:

1.
they should be continuous functions,
2.
they should be at least doubly differentiable,
3.
its square should be integrable,
4.
they should vanish at infinity (for finite systems),

When the Schrödinger equation is solved exactly (e.g., for the hydrogen atom), the resulting eigenfunctions tex2html_wrap_inline1687 form a complete set of functions, i.e., there is an infinite number of them, they are orthogonal to each other (or can be easily made orthogonal through a linear transformation), and any function which is of ``physical quality'' can be expressed through the combination of these eigenfunctions. Orthogonal means, that:

equation62

The eigenfunction tex2html_wrap_inline1691 corresponding to the lowest energy tex2html_wrap_inline1693 , describes the ground state of the system, and the higher energy values correspond to excited states. Physicists like to call tex2html_wrap_inline1695 's the states (since they contain all possible information about the state), while chemists usually call them wave functions.

Once the function tex2html_wrap_inline1695 (or its approximation, i.e., in case when the Schrödinger equation is solved only approximately) is known, the corresponding energy of the system can be calculated as an expectation value of the hamiltonian tex2html_wrap_inline1683 , as:

  equation67

where tex2html_wrap_inline1701 denotes the complex conjugate of tex2html_wrap_inline1695 since in general these functions may produce complex numbers. This is needed since the operator in this case represents a physical observable, and the result has to be a real number. This equation is frequently written using Dirac bra ( tex2html_wrap_inline1705 ) and ket ( tex2html_wrap_inline1707 ) notation to save space (and to confuse the innocent):

equation91

And if tex2html_wrap_inline1709 , i.e., the wave function is normalized, the equation looks even simpler:

equation95

  figure97
Figure 1: Volume element for a particle

Once we know the wave function tex2html_wrap_inline1695 for a given state of our system, we can calculate the expectation value for any quantity for which we can write down the operator. The wave function itself does not correspond to any physical quantity, but its square represents the probablity density. In other words:

equation101

or

equation109

or

equation120

represents the probablity that electron 1 is in the volume element tex2html_wrap_inline1713 around point tex2html_wrap_inline1677 , electron 2 is in the volume element of the size tex2html_wrap_inline1717 around point tex2html_wrap_inline1679 , and so on. If tex2html_wrap_inline1695 describes the system containing only a single electron, the tex2html_wrap_inline1723 simply represents the probability of finding an electron in the volume element of a size tex2html_wrap_inline1725 centered around point tex2html_wrap_inline1727 . If you use cartesian coordinates, then tex2html_wrap_inline1729 and the volume element would be a brick (rectangular parallelipiped) with dimensions tex2html_wrap_inline1731 whose vertex closes to the origine of coordinate system is located at (x, y, z). Now, if we integrate the function tex2html_wrap_inline1695 over all the space for all the variables (i.e., sum up the probablilities in all the elements tex2html_wrap_inline1737 ), we should get a probability of finding our electrons anywhere in the Universe, i.e., 1. This is why it is a good idea to normalize fundtion tex2html_wrap_inline1695 . If it is not normalized, it can easily be done by multiplying it by a normalization constant:

equation132

Since square of tex2html_wrap_inline1695 represents the probablility density of finding electrons, one may suspect, that it should be easy to calculate the total electron density from it. And actually it is:

equation140

where N is the total number of electrons, and tex2html_wrap_inline1745 is the famous Dirac delta function. In cartesians, it simply amounts to integrating over all electron positions vectors tex2html_wrap_inline1747 but one. Which one, is not important, since electrons are indistinguishable, and a proper wave function has to reflect this:

equation148

It is interesting to note, that for the wave function which describes the system containing only a single electron (but only then !!!):

  equation160

i.e., logically, the electron density and the probability density of finding the single electron are the same thing.

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Modified: Sun Mar 30 17:00:00 1997 GMT
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