How Accurately Does the Free Complement Wave Function of a Helium Atom Satisfy the Schrödinger Equation?
Physical Review Letters
The local energy defined by Hc =c must be equal to the exact energy E at any coordinate of an atom or molecule, as long as the c under consideration is exact. The discrepancy from E of this quantity is a stringent test of the accuracy of the calculated wave function. The H-square error for a normalized c , defined by 2 hc jðH À EÞ 2 jc i, is also a severe test of the accuracy. Using these quantities, we have examined the accuracy of our wave function of a helium atom calculated using the free
... ed using the free complement method that was developed to solve the Schrödinger equation. Together with the variational upper bound, the lower bound of the exact energy calculated using a modified Temple's formula ensured the definitely correct value of the helium fixed-nucleus ground state energy to be À2:903 724 377 034 119 598 311 159 245 194 4 a:u:, which is correct to 32 digits. The formulation of a general method for solving the Schrödinger equation (SE) is one of the most important subjects in quantum chemistry, although many scientists believed this important task to be impossible  . Recently, we have proposed the free iterative complement interaction (ICI) method [2-4], which we refer to in short as the free complement (FC) method, for solving the SE accurately in an analytical expansion form, and using this method, we have obtained very accurate energies and wave functions with excellent convergence for various systems     . In this communication, we examine and prove the "exactness" or the "high accuracy" of the calculated results using stringent theoretical tests of how well the calculated wave functions satisfy the SE. For such tests, we have examined the local energy, H-square error, and the upper and lower bounds of the exact energy. Exactness and accuracy of the wave function.-The SE, Hc ¼ Ec , is a local equation that must be satisfied at any coordinate. It can also be written as Hc ðrÞ c ðrÞ ¼ EðconstÞ ð8 rÞ; (1) where H is the Hamiltonian and c ðrÞ is the wave function at a coordinate r. The left-hand side of Eq. (1) is called local energy, E L ðrÞ, as E L ðrÞ Hc ðrÞ c ðrÞ : If c is not an exact wave function, then E L ðrÞ may depend on r. If E L ðrÞ is a constant at any point r, then Eq. (2) corresponds to Eq. (1), which is the SE. Therefore, the constancy of the local energy at any coordinate r is a straightforward test of how well the wave function c satisfies the SE. Another quantity that is useful to assess the exactness of a wave function is the H-square error 2 , defined by 2 hc jðH À EÞ 2 jc i; ( 3) for a normalized c where E ¼ hc jHjc i. The H-square equation we have utilized previously  corresponds to 2 ¼ 0, which is valid only for the exact wave function. The value of 2 is always positive and becomes zero only if c is exact. It is also related to the local energy by where hQi c 2 represents the expectation value of Q over the weight function jc j 2 . Thus, 2 is the variance of the local energy weighted by jc j 2 . Information on both the upper and lower bounds, E upper and E lower , of the exact energy E exact is very valuable to estimate the exact value of the energy. In this case, the exact energy is guaranteed to lie between E upper ! E exact ! E lower . When we use the variation principle, the calculated energy is the upper bound of the exact energy. There are several theories that produce the lower bound of the exact energy, and they are related to the H-square error 2 . One method is the Weinstein's lower bound energy E W lower , which is written as  This method is advantageous in that it only needs 2 and the energy expectation value. However, a problem of this method is that the quality (accuracy) of this lower bound is not good enough: it is too low usually to be useful. Another method was proposed by Temple , and is written as E T lower ¼ hc jHjc i À 2 E 1 À hc jHjc i : This method requires hc jHjc i, 2 , and in addition, the exact energy E