Energy to first order

Realizing that the strong field mixing resonances evolve from the extreme blue Stark states we can write their energies to first order in the electric field as... 

We consider the orbital energies to first order for a given value of t and write them as... 

This entire process is first-order perturbation theory. To get the total energy to first order, we add this correction to the zero-order energy ... 

Substituting this explicit expression for ri2 in Eq. 4.17 gives an integral we evaluate numerically, and it comes to 1.25 Ej, for He. For ground state He, the two-electron zero-order energy is Eq = —4.00 h> so the total energy to first-order is... 

Section 12-4 The Ground-State Energy to First-Order of Heliumlike Systems... 

The energy to first order is never below the exact ground-state energy. This is a general property of perturbation calculations as is easily proved (Problem 12-1). 

TABLE 12-1 Comparison of Exact Energy (in atomic units) with Energy to First Order when 7/ = r 2... 

EXAMPLE 12-5 Consider the methylene cyclopropene molecule, C4H4, in the HMO approximation. (See Appendix 6 for data.) At which carbon will a perturbation involving a affect the total n energy the most Calculate the energy to first order if the value of a at that atom increases to a -I- 0.1000 6. Calculate the energy change, to first order, of each of the four MOs. 

Figure 12-6 Energy to first order of n= 2 level of hydrogen as field strength (z-directed field).

Prove that the energy to first order for the lowest-energy state of a perturbed system is an upper bound for the exact energy of the lowest-energy state of the perturbed system, that is, that -h > W ),... 

Calculate the energy to first order of He in its lowest-energy state. Use the hydrogen atom in its ground state as your zeroth-order approximation. Use atomic units. Predict the signs (plus, zero, minus) of and g. Explain your reasoning. 

Calculate to first order the electronic energy of a hydrogen atom in its Is state and in the presence of an additional proton at a distance of 2 a.u. What is the total energy to first order Repeat for distances of 1 and 3 a.u. (See Appendix 3.)... 

Derive stationary-value conditions for the total energy obtained in Problem 8.4 from first principles, using the exponential operator variation that leads to (8.2.31). Hint Obtain the varied energy to first order and equate to zero the coefficients of (m virtual, i occupied).]... 

---