The key misconception you have is that the sum of the orbital energies is equal to the electronic energy. Compare the formulas for the two (taken from Chapter 3 of Szabo and Ostlund's Modern Quantum Chemistry)
If you simply sum up the orbital energies, you are double counting the electron-electron repulsion interactions.
For a closed-shell system, we can write the electronic energy in terms of the orbital energies, but we need additional terms:
To get Gaussian to print the core Hamiltonian (along with its components), you need to include the keyword
iop(3/33=5) if you also wanted to see the two electron integrals for some reason). You will also need
pop=full to print out the MO coefficients so you can transform the core Hamiltonian to MO basis.
For convenience, I'll just copy the results here:
0.58017 0.00000 1.42841
0.31436 0.78557 -0.94873
0.31436 -0.78557 -0.94873
-0.163989E+01 -0.107667E+01 -0.107667E+01
-0.107667E+01 -0.136306E+01 -0.377849E+00
-0.107667E+01 -0.377849E+00 -0.136306E+01
When this is done, we can now evaluate the Hartree-Fock Energy, which I did using Python:
>>> import numpy as np
>>> C = np.loadtxt("C.txt")
>>> H = np.loadtxt("H.txt")
>>> H_MO = C.T @ H @ C
>>> e_MO = -1.07569
>>> E_NR = 1.5516354770
>>> E_HF = H_MO[0,0] + e_MO + E_NR
This is in much better agreement and the only reason for the deviation is that the MO energies/coefficients and core Hamiltonian aren't printed to sufficient precision.
You can also get the Hartree-Fock energy by summing the individual components of the energy printed after the SCF, along with nuclear repulsion energy (requires
#p in the input line)
SCF Done: E(RHF) = -1.20555771776 A.U. after 4 cycles
NFock= 4 Conv=0.11D-09 -V/T= 2.1301
KE= 1.066737227463D+00 PE=-4.429751734382D+00 EE= 6.058213121791D-01
Where KE/PE/EE is the kinetic/potential/electronic energy.