Symmetrical orthogonalization

The Balanced orthogonalization is one of Per-Olov Löwdin developed (1916-2000), in quantum chemistry frequently used orthogonalization . As such, it serves to generate from a given non- orthogonal set of vectors an orthogonal set in which the scalar product is equal to zero for every two different vectors .

description

A basis is given for a subspace of a real or complex finite-dimensional vector space with scalar product ( or ). Let it be the matrix whose column vectors are the basis vectors of . ${\displaystyle S}$ ${\displaystyle V}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle A}$ ${\displaystyle S}$

Build the grief matrix . The Gram matrix is square , symmetric and positive definite (since the rows of are linearly independent and the scalar product is positive definite) and can thus be unitarily diagonalized . There is a unitary matrix and a diagonal matrix . ${\displaystyle A^{\dagger }A}$ ${\displaystyle A}$ ${\displaystyle U}$ ${\displaystyle D}$

${\displaystyle A^{\dagger }A=U^{\dagger }DU}$

and you can build the matrix . Then you build the matrix . The column vectors of form an orthonormal system because: ${\displaystyle H:=U^{\dagger }D^{-{\frac {1}{2}}}U}$ ${\displaystyle {\tilde {A}}:=AH}$ ${\displaystyle {\tilde {A}}}$

${\displaystyle {\tilde {A}}^{\dagger }{\tilde {A}}=(AH)^{\dagger }(AH)=(H^{\dagger }A^{\dagger })(AH)=(HA^{\dagger })(AH)=HH^{-2}H=Id}$

The columns of thus form the orthonormal basis of . ${\displaystyle {\tilde {A}}}$ ${\displaystyle V}$

Application in quantum chemistry

In quantum chemistry, the approximate , i.e. H. approximate solution of the electronic Schrödinger equation to generalized matrix eigenvalue problems of the form

${\displaystyle \mathbf {FC=SC\epsilon } }$,

with the Fock matrix , the coefficient matrix , which contains the LCAO coefficients of the molecular orbitals and the diagonal matrix of the orbital energies . To solve this eigenvalue problem, the equation is transformed to the so-called overlap matrix to the identity matrix is. With that the generalized eigenvalue problem would be reduced to an ordinary eigenvalue problem ${\displaystyle \mathbf {F} }$ ${\displaystyle \mathbf {C} }$ ${\displaystyle \mathbf {\epsilon } }$
${\displaystyle \mathbf {S} }$ ${\displaystyle \mathbf {E} }$

${\displaystyle \mathbf {F'C'=C'\epsilon } }$

reduced. For this purpose, the overlap matrix is diagonalized to the matrix by means of a unitary transformation , and then the roots of the reciprocal values ​​of the diagonal elements are drawn (yields ). Then the matrix is ​​"de-diagonalized" again by means of the inverse transformation. With the matrix obtained in this way and the relationship , the original equation can now be modified as follows: ${\displaystyle \mathbf {S} }$ ${\displaystyle \mathbf {s} }$ ${\displaystyle \mathbf {s} ^{-1/2}}$ ${\displaystyle \mathbf {X} }$ ${\displaystyle \mathbf {C=XC'} }$

${\displaystyle \mathbf {FXC'=SXC'\epsilon } }$.

By multiplying from the left side with the adjoint matrix we get: ${\displaystyle \mathbf {X^{\dagger }} }$

${\displaystyle \mathbf {X^{\dagger }FXC'=X^{\dagger }SXC'\epsilon } }$.

${\displaystyle \mathbf {X^{\dagger }SX} }$ but is just the identity matrix again, and we define

${\displaystyle \mathbf {F'=X^{\dagger }FX} }$.

With this we get the final result:

${\displaystyle \mathbf {F'C'=C'\epsilon } }$.

literature

• A. Szabo, NS Ostlund: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory . McGraw-Hill, 1989, ISBN 0-07-062739-8