首先介绍一下任意函数的Gram-Schmidt Orthonormalization操作，然后将其应用到高斯函数上

“Gram-Schmidt orthogonalization, also called the Gram-Schimdt process, is a procedure which takes a nonorthogonal set of linearly independent functions and constructs an orthongonal basis over an arbitary interval with respect to an arbitary weighting function w ( x ) w(x) w(x).” Given an original set of linearly independent functions { u n } n = 0 ∞ \{u_n\}_{n=0}^{\infty} {un​}n=0∞​, let { ψ n } n = 0 ∞ \{\psi_n\}_{n=0}^{\infty} {ψn​}n=0∞​ denote the orthogonalized (but not normalized) functions, { ϕ n } n = 0 ∞ \{\phi_n\}_{n=0}^{\infty} {ϕn​}n=0∞​ denote the orthornomarlized functions, and define ψ 0 ( x ) ≡ u 0 ( x ) ϕ 0 ( x ) ≡ ψ 0 ( x ) ϕ 0 2 ( x ) w ( x ) d x ϕ 1 ( x ) ≡ u 1 ( x ) + a 10 ϕ 0 ( x ) ∫ ϕ 0 2 ( x ) w d x = 1 a 10 = − ∫ u 1 ϕ 0 w d x ψ 1 = u 1 ( x ) − [ ∫ u 1 ϕ 0 w d x ] ϕ 0 ( x ) ϕ 1 ( x ) = ψ 1 ( x ) ϕ 1 2 ( x ) w ( x ) d x ϕ i ( x ) = ψ i ( x ) ϕ i 2 ( x ) w ( x ) d x ψ i ( x ) = u i + a i 0 ϕ 0 + a i 1 ϕ 1 + … + a i , i − 1 ϕ i − 1 a i j = − ∫ u i ϕ j w d x \psi_0(x)\equiv u_0(x) \\ \phi_0(x)\equiv {\psi_0(x)\over{\sqrt {\phi^2_0(x)w(x)dx}}} \\ \phi_1(x) \equiv u_1(x)+a_{10}\phi_0(x) \\ \int{\phi^2_0(x)}wdx=1 a_{10}=-\int u_1\phi_0wdx \\ \psi_1=u_1(x)-[\int u_1\phi_0wdx]\phi_0(x) \\ \phi_1(x)={\psi_1(x)\over{\sqrt {\phi^2_1(x)w(x)dx}}} \\ \phi_i(x)={\psi_i(x)\over{\sqrt {\phi^2_i(x)w(x)dx}}} \\ \psi_i(x)=u_i+a_{i0}\phi_0+a_{i1}\phi_1 +\ldots+ a_{i,i-1}\phi_{i-1} \\ a_{ij}=-\int u_i\phi_jwdx ψ0​(x)≡u0​(x)ϕ0​(x)≡ϕ02​(x)w(x)dx ​ψ0​(x)​ϕ1​(x)≡u1​(x)+a10​ϕ0​(x)∫ϕ02​(x)wdx=1a10​=−∫u1​ϕ0​wdxψ1​=u1​(x)−[∫u1​ϕ0​wdx]ϕ0​(x)ϕ1​(x)=ϕ12​(x)w(x)dx ​ψ1​(x)​ϕi​(x)=ϕi2​(x)w(x)dx ​ψi​(x)​ψi​(x)=ui​+ai0​ϕ0​+ai1​ϕ1​+…+ai,i−1​ϕi−1​aij​=−∫ui​ϕj​wdx If the functions are normalized to N j N_j Nj​ instead of 1, then ∫ a b [ ϕ j ( x ) ] 2 w d x = N j 2 ϕ j ( x ) = N i ψ i ( x ) ψ i 2 ( x ) w d x a i j = − ∫ u i ϕ j w d x N j 2 \int_{a}^{b}[\phi_j(x)]^2wdx=N_j^2 \\ \phi_j(x)=N_i{\psi_i(x)\over\sqrt{\psi^2_i(x)wdx}} a_{ij}=-{{\int u_i\phi_jwdx}\over{N^2_j}} ∫ab​[ϕj​(x)]2wdx=Nj2​ϕj​(x)=Ni​ψi2​(x)wdx ​ψi​(x)​aij​=−Nj2​∫ui​ϕj​wdx​

高斯基函数的径向函数为 R l ( r ) = r l e ( − α r 2 ) R_l(r)=r^{l}e^{(-\alpha r^2)} Rl​(r)=rle(−αr2), 积分区间为 r ∈ ( 0 , ∞ ) r\in(0,\infty) r∈(0,∞),weight functions是 w ( r ) = r 2 w(r)=r^2 w(r)=r2,下面以两个高斯基函数为例进行Gram-Schmidt Orthonormalization： u 0 ( r ) = r l e ( − α 1 r 2 ) u 1 ( r ) = r l e ( − α 2 r 2 ) w ( r ) = r 2 ψ 0 ( r ) ≡ u 0 ( r ) = r l e ( − α 1 r 2 ) ϕ o ( x ) ≡ ψ 0 ( x ) ψ 0 2 ( r ) w ( r ) d r = r l e ( − α 1 r 2 ) r 2 l e ( − 2 α 1 r 2 ) ⋅ r 2 d r u_0(r)=r^{l}e^{(-\alpha_1 r^2)}\\ u_1(r)=r^{l}e^{(-\alpha_2 r^2)}\\ w(r)=r^2\\ \psi_0(r)\equiv u_0(r) = r^{l}e^{(-\alpha_1 r^2)}\\ \phi_o(x)\equiv {\psi_0(x)\over{\sqrt {\psi^2_0(r)w(r)dr}}} \\ ={r^{l}e^{(-\alpha_1 r^2)}\over {\sqrt {r^{2l}e^{(-2\alpha_1 r^2)}\cdot r^2dr}}} \\ u0​(r)=rle(−α1​r2)u1​(r)=rle(−α2​r2)w(r)=r2ψ0​(r)≡u0​(r)=rle(−α1​r2)ϕo​(x)≡ψ02​(r)w(r)dr ​ψ0​(x)​=r2le(−2α1​r2)⋅r2dr ​rle(−α1​r2)​