= Linear combination of atomic orbitals
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= LCAO
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The <LCAO> approximation builds a <molecular orbital> as a <linear combination> of single-centre <atomic orbitals>. For two equivalent normalized real $1s$ functions $\phi_A$ and $\phi_B$, choose their signs positive and define their <orbital overlap> $S$. At nonzero separation, $0<S<1$, and the normalized combinations are
$$
\sigma_\pm(\mathbf r)=\frac{\phi_A(\mathbf r)\pm\phi_B(\mathbf r)}{\sqrt{2(1\pm S)}},\qquad
S=\int_{\mathbb R^3}\phi_A(\mathbf r)\phi_B(\mathbf r)\,d^3\mathbf r.
$$
Here $\sigma_+=\sigma_g$ and $\sigma_-=\sigma_u^*$. Equal coefficients use the equivalence of the two centres; a general <LCAO> need not have equal coefficients. See https://ocw.mit.edu/courses/5-61-physical-chemistry-fall-2017/ef9da1132ff6233d732a1d6115099467_MIT5_61F17_lec24.pdf[Field, MIT 5.61, lecture 24, pp. 3–6: LCAO coefficients and overlap integrals].
For the <Gaussian function> toy's $a=1$, the two denominators are $\sqrt{2(1+S)}\approx1.654013$ and $\sqrt{2(1-S)}\approx1.124385$. To check <wavefunction normalization>, expand the square: $\int(\phi_A\pm\phi_B)^2=1+1\pm2S$. Dividing by $2(1\pm S)$ gives one. The local <probability density> is
$$
|\sigma_\pm|^2=\frac{\phi_A^2+\phi_B^2\pm2\phi_A\phi_B}{2(1\pm S)}.
$$
The cross term is the <interference>; adding the separate densities first would lose it. These formulas normalize the chosen <linear combination> exactly, while the restricted choice of <atomic orbitals> remains an approximation to an actual <molecular orbital>.
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