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Let's look at scheme RHF

Do I understand correctly, the origin of Virtual Orbitals is a solution of (for, exampre, restricted) HF equation eigenfuncions?

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Yes, they result from the HF equations just like the occupied orbitals. The diagram should also be show them as present in the initial set of equations, unless the basis (used for the LCAO) was changed between the two states shown. The number of occupied orbitals plus the number of virtual orbitals is equal to the basis set size because they are the eigensolutions of $$ \mathbf{F}\mathbf{C} = \mathbf{\epsilon}\mathbf{S}\mathbf{C}, $$ where $\mathbf{F}$ is the Fock matrix, $\mathbf{C}$ is the matrix of orbital coefficients, $\mathbf{\epsilon}$ are the eigenvalues which are the orbital energies, and $\mathbf{S}$ is the atomic orbital overlap matrix (because the basis set used is typically not orthogonal). The dimension of all matrices involved is determined by the size of the basis set, which will be finite because otherwise the computational effort becomes infinite.

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  • $\begingroup$ Can you, please, clarify (may be, it is need some math) why the number of occupied orbitals plus the number of virtual orbitals is equal to the basis set size, because I thaught the sum is infinite, as any problem for eigenvalues and eigenfunctions. $\endgroup$ – Sergio Dec 13 '20 at 11:36
  • $\begingroup$ @Sergio I have extended my answer. $\endgroup$ – TAR86 Dec 13 '20 at 13:26
  • $\begingroup$ It seems, I understend.The virtual orbital in H2 molecule is a unbonding orbital. Am I right? $\endgroup$ – Sergio Dec 13 '20 at 16:55
  • $\begingroup$ That depends on the basis set. Assuming the minimal basis set of two hydrogen-like orbitals, you are correct. $\endgroup$ – TAR86 Dec 13 '20 at 17:48
  • $\begingroup$ Yes, I mean minimal basis set. Are there virtual orbitals if the HF method is applied to atoms? Or this is a feature of HF with MO LCAO? $\endgroup$ – Sergio Dec 13 '20 at 17:52

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