第二章 密度泛函理论

If
If the basis set is continuous, the orthonormality condition becomes
An operator transforms a ket into another ket in the Hilbert space, The adjoint of ,,denoted by , ,transforms the corresponding bra, For normalized ket and bra,( 2.1.11) and (2.1.12)can be written
If the representation is continuous, one has an integral rather than a sum, for example,
The inner product of a ket and a bra is a complex number and satisfies
Several tools are developed for formal :
Dirac notation Density operators Density matrices
We begin with the quantum state of a single-particle system. Such a state was described in Chapter 1 by a wave function (r) in coordinate space (neglecting the spin for the moment). It can also beequivalently "represented" by a momentumspace wave function that is the Fourier transform of (r) .
A system in a mixed state can be characterized by a probability Biblioteka Baiduistribution over all the accessible pure states. To accomplish this description, we generalize the density operator of( 2.2.3) to the ensemble density operator
Projection operators have the property: For this reason, they are said to be idempotent.
By inserting (2.1.7) into (2.1.6), we get
From which follows where / is the identity operator. This is the closure relation. The corresponding expression for a continuous basis set is
while for fermions, a typical normalized antisymmetric basis ket would be
The closure relation in
is
while that in
is
2.2 Density operators
since
Note that is a projection operator. We then have for normalized
Chapter 2 DENSITY MATRICES
2.1 Description of quantum states and the Dirac notation 2.2 Density operators 2.3 Reduced density matrices for fermion systems 2.4 Spinless density matrices 2.5 Hartree-Fock theory in density-matrix form 2.6 The N-representability of reduced density matrices 2.7 Statistical mechanics
Note that if the interactions can induce change in particle number, the accessible states can involve different particle numbers.
For a system in a pure state, one is 1 and the rest are zero; of (2.2.7) then reduces to of (2.2.3). By construction, is normalized: In an arbitrary complete basis ,
The closure relation greatly facilitates transformation between different representations, which makes the Dirac notation so useful. As an example, we compute the inner product
The linearity of the Hilbert space implements the superposition principle: a linear combination of two vectors Cx |W,) + C2|V2) is also a ket vector in the same Hilbert space, associated with a realizable physical state.
If particle spin is included in the above, then the closure relation is We now turn to a quantum system of many identical particles, for which the foregoing concepts and formulas go through when suitably generalized. However, a new feature appears—the antisymmetry （or symmetry）of fermion(or boson) wave functions with respect to exchange of indices
This, together with the quantum superposition principle, leads one to construct a more general and abstract form of quantum mechanics. Thus, one associates with each state a ket vector I in the linear vector space , called the Hilbert space .
A very important type of operator is the projection operator onto a normalized ket
The projection property is manifest when Pt acts on the ket |W) of (2.1.6):
is Hermitian:
It also is positive semidefinite: The are the eigenvalues of .
2.1 Description of quantum states and the Dirac notation
In this chapter, the concepts and form of elementary quantum mechanics are generalized. This allows use of variables other than coordinates for the description of a state, permits ready discussion of physical states that cannot be described by wave functions, and prepares the way for formally considering the number of particles to be variable rather than constant.
An operator description of a quantum state becomes necessary when the state cannot be represented by a linear superposition of eigenstates of a particular Hamiltonian .This occurs when the system of interest is part of a larger closed system, as for example an individual electron in a manyelectron system, For such a system one does not have a complete Hamiltonian containing only its own degrees of freedom, thereby precluding the wave-function description. A state is said to be pure if it is described by a wave function, mixed if it cannot be described by a wave function.
which is identically (2.1.1). Or, consider the effect of the operator in (2.1.11),
If we use a continuous basis set, (2.1.22) becomes
As another example of the use of (2.1.15), we may prove the formula for the decomposition of a Hermitian operator into its eigenfunctions. Let the kets be the complete set of eigenkets of the linear operator A ,with eigenvalues . Then
(coordinates) of any two particles. The antisymmetric and symmetric states span subspaces of the N-particle Hilbert space, , the subspaces denoted by and . We focus on , since electrons are fermions. In , a normalized basis ket for N particles in suitably defined states \<X\), \oc2),..., \ocN), respectively, is