Lower Bounds for Nonrelativistic Atomic Energies

A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput.2021, 17, 1535) is further developed and applied to the highly accurate calculation of the ground-state energy of two- (He, Li+, and H–) and three- (Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.


Supporting Information Derivation of Matrix Elements
The 1 r ij rpa expectation value, corresponding to the product of the electron-electron and electron-nuclei potential energy operators, can be solved by beginning with Eq. 3.1, ψ(r, σ) = A{φ(r, A)χ(σ, θ)} . (S1) The spin functions are constructed by coupling elementary spin functions to the selected total spin quantum numbers. As such, it is the spatial ECG functions φ(r, A) that are of interest when deriving the expectation value, where the kth ECG basis function is written r ∈ R 3np is the vector containing the Cartesian positions of the n p particles, A k ∈ R np×np is the positive-definite symmetric matrix containing the non-linear parameters of the kth ECG basis function and I 3 ∈ R 3×3 is the identity matrix. Then, the kth, lth element takes the where we have defined A kl = (A k ⊗ I 3 )+(A l ⊗ I 3 ). By using the following Gaussian integral transforms, we arrive at where J mn ∈ R np×np is a rank-1 matrix with element Furthermore, we define J mn = J mn ⊗ I 3 . Using the Gaussian integral expression for n-electrons, we obtain The determinant above can be rewritten an meaning we can use the following rank-2 determinant expression, where rank(H 1 ) = rank(H 2 ) = 1.
For matrices A and B, it can be shown that rank (AB) = rank (A), if B is a non-singular matrix. As such, rank J ij A −1 kl = rank (J ij ). As each column of J ij is a linear combination of the first, it is a rank-1 matrix, meaning that the J ij A −1 kl matrix is also rank-1. The same is true for J pa A −1 kl .
By letting H 1 = t 2 J ij A −1 kl and H 2 = u 2 J pa A −1 kl , the expectation value then reads where we have defined We can then see that which gives (S16) The following substitution can then be made, leading to (S18) By making the substitution the expectation value then can be written as Hence, evaluation of Eq. S21 leads to the final expression for ab − c > 0, c = 0.
This expression can be generalized to the 1 expectation value that appears in the V 2 ee operator of the 2-electron case. Then, the J pa matrix can be seen to be equal to the J ij matrix. The traces of Eqs. S12-S14 are such that a = b and c = a 2 , where we have used the fact that for a rank-1 matrix M Equation S22 then reads as As discussed in the paper, an essential part of the PM lower bound theory is the assurance of the lower bound property of the poles of the PM equation. For this purpose, we consider two strategies. The first one, used in Ref. 1, is to check whether x j decreases monotonically when increasing the dimensionality of the basis set. This test is self-contained, but not directly applicable to the current setup with ECGs. This approach is useful if the basis set is increased "systematically" without changing the already selected functions, e.g., as it is in the case of increasing the dimensionality of basis sets composed of orthogonal polynomials.
The ECG basis set including functions that are generated and regularly refined based on the energy-minimization condition does not necessarily have this property.
The second strategy is testing the fulfillment of the x j (ε 1 ) ≥ ε j+1 condition that requires information about ground and excited-state energies (that are generally not known). In particular, for Li, we know that we have a good upper bound from the ECG computation that can be estimated to be converged on the order of a few ppbs. Then, we may take this value and lower it by 50 ppb to have a lower-bound estimate. Then, by using this value for ε in the PM equation one finds the pole x j and checks whether it is greater than ε j+1 (that is also taken from some other computation). Is this a "useful" check? In its present form, it requires external information that is, in general, not available. There is work underway, starting out from this particular condition that we check here, to eliminate the need to use information on state energies, if the overall computation is sufficiently accurate.
The results of this (second) test are shown for the Li atom using the energy-optimized basis set in Table S1. The column denoted by is also plotted in Fig. 1d of the paper. Table S1: Li atom ground state energy lower bound: testing the validity of the x 1 (ε 1 ) ≥ ε 2 condition which assures that the PM expression gives a lower bound to the ground state of the Li atom when replacing x 1 with ε 2 in the PM equation. The PM matrix was calculated using the Ritz eigenvectors corresponding to an energy-optimized subspace. (See Table 2 in the paper, for the ε n values.) Similar results are shown in Table S2 for the variance-optimized Lehmann basis set results. Here, the fourth column denoted by x 2 (ε 1 )[E h ] is the check whether the second root of the PM equation is greater than or equal to ε 3 for which the precise value is given in the last row of the table. One notes that for all basis set sizes, L, the condition x 2 (ε 1 ) ≥ ε 3 is fulfilled. The last column shows the Lehmann lower bound values that are not accurate in this case. Table S2: Li atom ground-state energy lower bound: Test for the validity of the condition The test is shown for results based on a variance-optimized Lehmann eigenvector basis set with ρ = ε = ε − 3 , a lower bound to the second excited state (See Table 2 for the ε n values used here). Further explanation of the data is provided in the text. (last column), generated using the pole ρ = ε − 2 . Table S3: He atom ground state energy lower bound: Test for the validity of the condition that x 1 (ε 1 ) ≥ ε 2 for the He atom for different variance-optimized basis sets and Lehmann eigenvectors. The notation is as in Tables S1 and S2. (See Table 2 for the ε n values used.) The last column gives the Lehmann lower bounds.