1 \part{Mathematical foundations}
3 Reminder: Modern Quantum Chemistry \& Sakurai \& Group Theory \ldots
5 \chapter{Linear algebra}
7 \section{Vectors and bases}
9 A vector $\vec{a}$ of an $N$-dimensional vector space (see \ref{math_app:vector_space} for mathematical details) is represented by its components $a_i$ with respect to a set of $N$ basis vectors ${\vec{e}_i}$.
11 \vec{a}=\sum_i^N \vec{e}_i a_i
14 The scalar product for an $N$-dimensional real vector space is defined as
16 (\vec{a},\vec{b})=\sum_i^N a_i b_i \text{ ,}
19 which enables to define a norm
21 ||\vec{a}||=\sqrt{(\vec{a},\vec{a})}
23 that just corresponds to the length of vector \vec{a}.
24 Evaluating the scalar product $(\vec{a},\vec{b})$ by the sum representation of \eqref{eq:vec_sum} leads to
26 (\vec{a},\vec{b})=(\sum_i\vec{e}_ia_i,\sum_j\vec{e}_jb_j)=
27 \sum_i\sum_j(\vec{e}_i,\vec{e}_j)a_ib_j \text { ,}
29 which is equal to \eqref{eq:vec_sp} only if
31 (\vec{e}_i,\vec{e}_j)=
32 \delta_{ij} = \left\{ \begin{array}{lll}
33 0 & {\rm for} ~i \neq j \\
34 1 & {\rm for} ~i = j \end{array} \right.
35 \text{ (Kronecker delta symbol),}
37 i.e.\ the basis vectors are mutually perpendicular (orthogonal) and have unit length (normalized).
38 Such a basis set is called orthonormal.
39 The component of a vector can be obtained by taking the scalar product with the respective basis vector.
41 (\vec{e}_j,\vec{a})=(\vec{e}_j,\sum_i \vec{e}_ia_i)=
42 \sum_i (\vec{e}_j,\vec{e}_i)a_i=
43 \sum_i\delta_{ij}a_i=a_j
45 Inserting the expression for the coefficients into \eqref{eq:vec_sum}, the vector can be written as
48 \vec{a}=\sum_i \vec{e}_i (\vec{e}_i,\vec{a}) \Leftrightarrow
49 \sum_i\vec{e}_i\cdot \vec{e}_i=\vec{1}
51 if the basis is complete.
52 Indeed, the very important identity representation by the outer product ($\cdot$) in the second part of \eqref{eq:complete} is known as the completeness relation or closure.