Dirac notation for integrals

Dirac Notation for Integrals Energy Functional to Be Minimized Energy Minimization with Constraints Slater Determinant Subject to a Unitary Transformation The J and K Operators Are Invariant Diagonalization of the Lagrange Multipliers... 

Paul Dirac introduced some useful formal tools (such as his notation for integrals and operators). One of them is the Dirac delta function S (x), an object then unknown to mathematicians, which mrned out to be very useful in physics. We may think of it as of a function with the following characteristics ... 

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

The bracket (bra-c-ket) in

) provides the names for the component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket tp) with its corresponding bra (-01 gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra tpj and the ket Aj>i) is expressed in Dirac notation as (0yjA 0,) or as J A i). These scalar products are also known as the matrix elements of A and are sometimes denoted by Ay. 

Within the old adiabatic approximation, Eq. (39) is the basic starting point. However, from here on, the various approximations diverge. For ease of discussion, we shall first still make the Condon approximation, and then give the further approximations. However, it must be kept in mind that many similar approximations are also made in papers that use a non-Condon approach. The basic premise of the Condon approximation is that the electronic part of the matrix element varies sufficiently slowly with Q so that it can be taken out of the integration over dQ. The matrix element then reduces to products of electronic and vibrational integrals. In Dirac notation... 

Here pj7m) denotes the spherical-wave free-electron function with the usual notations for Dirac angular quantum numbers. The numbers jlm are fixed by the overlap with the bound-electron wave function a) = njlm) where n is the principal quantum number. Integration over p is interpreted as integration over energies Ep = /p2 + m2. 

In the Dirac notation, quantum mechanical integrals are represented by a shorthand notation using brackets, for instance ... 

Here we have introduced the bra ((V ) and ket i ic)) notation of Dirac which for us will serve as a shorthand way of representing certain integrals. This assertion is of special importance in our enterprise in the context of the energy for which the relevant expectation value may be written... 

The integrals over the spatial and spin coordinates (0 are the spinorbitals, tp - the orbitals) in the Dirac notation wUl be denoted with angle brackets () h denotes a one-electron operator and r 2 - the distance between electrons 1 and 2), as follows for the one-electron integrals. 

Here, we have also switched to the bra and ket notation that was introduced by Dirac to deal with wavefunctions in an elegantly compact manner. In Dirac notation, a wavefiinction is denoted by a ket i.e. the Is orbital has been written 1 ) and the corresponding complex conjugate would be shown as (. If a bra and ket appear in the right order to complete a bracket , then integration is implied for example ... 

Remembering that for antisymmetrized two-electron integrals in Dirac s notation, pqWrs) = - pqWsr) = - ( pllrs) = qpWsr), we may re-index sums and combine terms where appropriate to obtain... 

---