We present an elementary pedagogical derivation of the Brillouin-Wigner and the Rayleigh-Schrödinger perturbation theories with Epstein-Nesbet partitioning. A variant of the Brillouin-Wigner perturbation theory is also introduced, which can be easily extended to the quasi-degenerate case. A main advantage of the new theory is that the computing time required for obtaining the successive higher-order results is minimal after the third-order calculation. We illustrate the accuracy of the new perturbation theory for some simple model systems like the perturbed harmonic oscillator and the particle in a box.
One of the most useful approximation methods in quan-tum mechanics is the perturbation theory. 1 There are many variations of the perturbation theory for dealing with station-ary bound systems. Of these, the Rayleigh-Schrödinger per-turbation theory (RSPT) and the Brillouin-Wigner pertur-bation theory (BWPT) are the most fundamental and general ones. 2 3 Some theories may give more accurate results but their applicability is limited to specific systems. In this paper, we present an elementary derivation of BWPT and RSPT with Epstein-Nesbet (EN) partitioning, 4 5 and introduce a new variant of the BWPT. A major advantage of the new perturbation theory is that the computing time requir-ed for obtaining the successive higher-order results beyond the third-order calculation is negligible compared to that required for the calculations of the second and the third order results. For simple model systems like the perturbed harmonic oscillator and the particle in a box, it is shown that the ground-state energies calculated from the new perturbation theory have comparable accuracy as those calculated up to the same order by BWPT and RSPT with EN partitioning.
Note that Eqs. (6) and (7) are exact relations. In the RSPT, it is assumed that the energy and the state vector of the perturbed system are analytic functions, so that they can be represented as MacLaurin series in λ . In Eq. (7), no such assumption is made, but it involves the unknown quantity En .
The k th-order energy of the BWPT is given by truncating the series in Eq. (9) after the λk term. The resulting equation is solved for En to give the k th-order energy E n (BWk) ; in this paper E n (BWk) denotes an approximate energy that is correct up to O ( λ k ), not the k th-order correction. A more approxi-mate procedure to evaluate E n (BWk) is just to put the value En (BW,k–1) for E n in the truncated series on the right hand side.
These expressions coincide with those of the RSPT in Eqs. (11) and (12). With Eqs. (19) and (20) it is straightforward to calculate the higher order quantities by using the recursive relations in Eqs. (17) and (18). One can see easily that in this new perturbation method all the matrix elements of the perturbation Hamiltonian are calculated at the level of third-order energy calculation, so that the computing time requir-ed for obtaining the successive higher-order results beyond the third-order calculation is minimal.
where we will assume that | φ i (0) 〉 is orthonormalized. The n th state of interest, | Ψ n 〉 , corresponds to the correct zeroth-order state | φ n (0) 〉 , for which the expansion coefficient c nn (0) has the larger absolute value than other expansion coefficients.
The second-order energy expression in Eq. (35) coincide with that of the multireference perturbation theory of Chen, Davidson, and Iwata 6 Eq. (20) of Ref. 6 truncated at the second order]. Multireference perturbation theories based on the RSPT and the BWPT with various partitioning of the Hamiltonian matrix have been widely used in molecular electronic problems. 3 5 7 - 10 We will show below that Eq. (35) provides quite accurate results by including just a few nearby states in constructing the correct zeroth-order states.
Although this example must be trivial, it serves well to illustrate the quality of the approximations. The exact energy is given by Enexact =(1+ λ ) 1/2 ħω ( n +1/2). In Figure 1(a) we display the relative errors in the ground-state energies E 0 (k) calculated from Eq. (18). We set the values of , ħ and ω to unity. For clarity's sake, we give only the even order results up to E 0 (6) . In fact, E 0 (2k+1) is slightly worse than E 0 (2k) , though better than E 0 (2k–2) . Numerical results up to k = 20 and λ = 50 show that E 0 (k) ( λ ) and E 1 ( k )( λ ) converge to the exact values as we go to the higher order; these two states are the lowest energy states having the even and the odd parity, respectively. However, as the quantum number n increases, the radius of convergence appears to decrease.
Figures 1(b) and 1(c) display the corresponding results calculated from the RSPT and the BWPT with EN partition-ing, respectively. We see that both perturbation theories also provide convergent results. For this model system, the RSPT gives more accurate results than the BWPT, and the results calculated from our perturbation theory lie between those from the two theories. On the other hand, as shown in Figure 1(d) , the RSPT and the BWPT without EN partitioning give divergent results even for very small λ .
In Figure 1(a) , we also compare the results of Eq. (35) (drawn as the black dashed curve), which utilizes the correct zeroth-order states, with those obtained from Eq. (18). To calculate the ground state energy, we just mix the | Ψ 0 (0) 〉 and | Ψ 2 (0) 〉 states to obtain | φ 0 (0) 〉 and | φ 2 (0) 〉 states. Even with this small tweaking, the resulting second-order energy is almost as good as the straightforward fourth-order energy obtained from Eq. (18).
Relative errors in the ground-state energies of the harmonically perturbed oscillator [Eq. (36)] calculated from the various perturbation theories as a function of the perturbation parameter λ. Figures 1(a), 1(b), and 1(c) display the results of the present, RS, and BW perturbation theories with EN partitioning, while Fig. 1(d) shows the results of the RS and BW perturbation theories without EN partitioning.
Figure 2 displays the results for a modified particle in a box with infinite walls at x = − L /2 and x = L /2. Inside the box, the potential energy is given by V ( x ) = λ cos( πx / L ). An interesting feature of this system is that when λ has a large positive value, V ( x ) becomes a deep double-well potential and the ground state and the first excited state become almost degenerate, and the energy difference between the ground state and the second excited state gets small. See Figure 2(a) for the variation of numerically calculated energy levels as a function of λ ; in the calculation, the values of ħ , μ and L were set to unity. Hence the second-order energy calculated from Eq. (18) may become inaccurate for large positive λ values, as shown in Figure 2(b) . In com-parison, E 0 (2) calculated from Eq. (35) is as accurate as the fourth-order energy calculated from Eq. (18) for the whole range of λ considered; in applying Eq. (35) | ψ 1 (0) 〉 and | ψ 3 (0) 〉 states are mixed to obtain | ϕ 1 (0) 〉 and | ϕ 3 (0) 〉 . Note that the downward spikes near λ = 0 result from the fact that the ground-state energy becomes zero when λ has a small negative value.
(a) Variation of the numerical Enexact(λ) values for a modified particle in a box with V(x) = λ cos(πx/L). (b)-(e) Com-parison of the relative errors in the ground-state energies calcu-lated from various perturbation theories. Figures 2(b), 2(c), and 2(d) display the results of the present, RS, and BW perturbation theories with EN partitioning, while Figure 2(e) shows the results of the RS and BW perturbation theories without EN partitioning.
Figures 2(c) and 2(d) display the corresponding results calculated from the RSPT and the BWPT with EN partition-ing, respectively. For this model system, the results of our perturbation theory and the BWPT show better convergence behavior than those of the RSPT with EN partitioning. Again, as shown in Figure 2(e) , the RSPT and the BWPT without EN partitioning give much less accurate results; note that the ordinate scale in Figure 2(e) is different from those of Figure 2(b) , 2(c) , and 2(d) .
We have shown that various perturbation theories with EN partitioning can be derived in a quite elementary manner. In particular, we proposed a variant of the BWPT, which can be easily extended to the quasi-degenerate case. While the usual BWPT requires rapidly increasing computing time to get the higher-order results, the new perturbation theory requires negligible computing time to get the successive higher-order results beyond the third-order calculation. At least for the two simple model systems considered, the new perturbation theory appears to give as accurate results as the BWPT and the RSPT with EN partitioning. The new variant BWPT would carry the similar advantages and disadvantages of the usual BWPT in comparison with the RSPT. 3 In parti-cular, it does not satisfy the size-consistency requirement, so that it might have a limited usability in molecular electronic problems.
The authors would like to thank Prof. Haruyuki Nakano for helpful comments. This work was supported by the grants from National Research Foundation (NRF), funded by the Korean Government (Grant Nos. 2012-R1A1A2003055 and 2010-0001631).
 S Lee , CH Choi , E Kim , and YK Choi , "A Variant of the Brillouin-Wigner Perturbation Theory with Epstein-Nesbet Partitioning", Bulletin of the Korean Chemical Society, vol. 11, no. 11, Nov 2013.