Appendix
A.1 Vectors and Matrices Spaces and Operators
|ψ and ψ| (with the former being a row matrix of ψ values and the latter a column
matrix of its complex conjugated ψ
∗ ) denote the “bra" and “ket" vectors of the braket
space of popular use in quantum chemistry. Symbols denoting matrices are written
in bold and the determinant of matrix A is written as |A| or det (A). The transposed
matrix A is written as A
T , the adjugate matrix A is written as A
+ , and the inverse
matrix A is written as A
−1 . The unit matrix (1) is the matrix made of all zeroes but
diagonal ones which have value 1, while the matrix made of all zeroes is the zero
matrix 0.
A vector space of dimension n is called V
n with V
2 corresponding to a plane (say
x,y) and V
3 corresponding to a three-dimensional space (say x, y, z). Basis sets of the
vector space V
n are usually denoted as {e i }
n . In the case of V
3 , one has usually {e i }
3 =
(|i, | j, |k). The Function space F
n is the analogue of the vector space consisting of
n linearly independent basis functions φ i ({φ i }
n ) owing to the fact that one replaces the
variable x with its function φ i (strictly speaking this is a continuous transformation
but it can be considered as the limit x → 0 of the function representation in
steps of x corresponding to V
∞ ). This vector–function equivalence allows easier
manipulations of functions especially for compute purposes.
Vectors and functions are often constructed from sets of basis vectors or functions
like the above-mentioned {e i }
n and {φ i }
n . Operators acting on vectors and matrices
are marked by the “hat." Special operators are the direct sum (⊕) and product (⊗).
Of particular interest for quantum chemistry are the Nabla (∇) defined as
∂
∂x
|i +
∂
∂ y
| j +
∂
∂z
|k,
(A.1)
and the Laplacian ( or ∇
2 ) defined as
∂
2
∂x 2 +
∂
2
∂ y 2 +
∂
2
∂x 2 .
(A.2)
© Springer International Publishing AG 2018
A. Laganà and G. A. Parker (eds.), Chemical Reactions, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-62356-6
197
A.1 Vectors and Matrices Spaces and Operators
|ψ and ψ| (with the former being a row matrix of ψ values and the latter a column
matrix of its complex conjugated ψ
∗ ) denote the “bra" and “ket" vectors of the braket
space of popular use in quantum chemistry. Symbols denoting matrices are written
in bold and the determinant of matrix A is written as |A| or det (A). The transposed
matrix A is written as A
T , the adjugate matrix A is written as A
+ , and the inverse
matrix A is written as A
−1 . The unit matrix (1) is the matrix made of all zeroes but
diagonal ones which have value 1, while the matrix made of all zeroes is the zero
matrix 0.
A vector space of dimension n is called V
n with V
2 corresponding to a plane (say
x,y) and V
3 corresponding to a three-dimensional space (say x, y, z). Basis sets of the
vector space V
n are usually denoted as {e i }
n . In the case of V
3 , one has usually {e i }
3 =
(|i, | j, |k). The Function space F
n is the analogue of the vector space consisting of
n linearly independent basis functions φ i ({φ i }
n ) owing to the fact that one replaces the
variable x with its function φ i (strictly speaking this is a continuous transformation
but it can be considered as the limit x → 0 of the function representation in
steps of x corresponding to V
∞ ). This vector–function equivalence allows easier
manipulations of functions especially for compute purposes.
Vectors and functions are often constructed from sets of basis vectors or functions
like the above-mentioned {e i }
n and {φ i }
n . Operators acting on vectors and matrices
are marked by the “hat." Special operators are the direct sum (⊕) and product (⊗).
Of particular interest for quantum chemistry are the Nabla (∇) defined as
∂
∂x
|i +
∂
∂ y
| j +
∂
∂z
|k,
(A.1)
and the Laplacian ( or ∇
2 ) defined as
∂
2
∂x 2 +
∂
2
∂ y 2 +
∂
2
∂x 2 .
(A.2)
© Springer International Publishing AG 2018
A. Laganà and G. A. Parker (eds.), Chemical Reactions, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-62356-6
197