Do bonds between the atoms influence ab-initio energies?

As you know, oxygen has two available states. However, the picture above illustrates oxygen (red) has a just one bonding with one carbon. I am kind of afraid it makes differences compared that an oxygen has two bondings with another carbon. To my knowledge bonding is nothing but overlapping "probability clouds" to find electrons between two atoms. Bonding is just chemical expression to see it intuitively.

Do you think my result is different with the one has additional bonding with another carbon atom?

The bonds you draw or see in Gaussview should have no effect on the calculation (besides some minor effects in choosing internal coordinates for a geometry optimization). As you note, a bond is a chemical concept rather than a quantum mechanical one; an electronic structure calculation depends on the molecular Hamiltonian (or some approximation of it) which doesn't include any notion of individual bonds, just charges and distances between them.

Programs like Gaussview try to give a chemists view of a molecule by drawing bonds. The way it decides what is bonded generally isn't too sophisticated; usually a bond is drawn if a pair of elements A-B is closer than some threshold value chosen to represent a typical bond for those two species.

Adding to the (correct) answer by Tyberius, I'll just point out that ab initio calculations of electronic energies of a molecule depend on a small number of things, most notably:

• (1) spin multiplicity,
• (2) charge (or total number of electrons),
• (3) geometry.

The two most popular ways to describe the geometry geometry are:

(A) The XYZ format (Cartesian coordinates):

C     0.000000     0.000000     0.000000
H     0.000000     0.000000     1.089000
H     1.026719     0.000000    -0.363000
H    -0.513360    -0.889165    -0.363000
H    -0.513360     0.889165    -0.363000


(B) The Z-Matrix format:

C
H   1 1.089000
H   1 1.089000  2  109.4710
H   1 1.089000  2  109.4710  3  120.0000
H   1 1.089000  2  109.4710  3 -120.0000


Notice that in both the XYZ format and the ZMat format, we just give the (x,y,z) positions of the nuclei (in the XYZ case) or the bond lengths and angles (in the ZMat case) without any mention at all about what is "bonded". The idea is that the quantum chemistry calculation itself will figure out the "bonding" based on just the geometry, charge and spin multiplicity.