We investigate the structure of the exact wave function as a solution of the Schrödinger equation, aiming the singles and doubles description of the exact wave function. The basis is that the Hamiltonian involves only one and two body operators. We first present two theorems that indicate a possibility of the singles and doubles description of the exact wave function. We then examine the exponential ansatz, as this theorem implies it to be a possible structure of the exact wave function. Variational CCS (singles) wave function is shown to be certainly exact for one particle Hamiltonian. Thouless transformation plays an important role in the formulation. The conventional CCSD (singles and doubles) function is restrictive, even if it is solved variationally. A wider coupled cluster function with general singles and doubles substitution operators (CCGSD) is also not exact for the existence of noncommuting operators. We then analyze some formal properties of the full CI wave function, and finally, we propose an ansatz of the exact wave function and describe the method of solution. It involves successive solutions of the secular equations of the order of singles and doubles. It is variational and we can calculate both ground and excited states.
We investigate the structure of the exact wave function as a solution of the Schrödinger equation, aiming the singles and doubles description of the exact wave function. The basis is that the Hamiltonian involves only one and two body operators. We first present two theorems that indicate a possibility of the singles and doubles description of the exact wave function. We then examine the exponential ansatz, as this theorem implies it to be a possible structure of the exact wave function. Variational CCS (singles) wave function is shown to be certainly exact for one particle Hamiltonian. Thouless transformation plays an important role in the formulation. The conventional CCSD (singles and doubles) function is restrictive, even if it is solved variationally. A wider coupled cluster function with general singles and doubles substitution operators (CCGSD) is also not exact for the existence of noncommuting operators. We then analyze some formal properties of the full CI wave function, and finally, we propose an ansatz of the exact wave function and describe the method of solution. It involves successive solutions of the secular equations of the order of singles and doubles. It is variational and we can calculate both ground and excited states.
We propose an explicit factorized formula for the ground state many-body wave function of a model quartic potential problem in two and three dimensions, using our recently developed method for solving Schrödinger equations. Our novel formula has a clear real-space structure: it is exactly factorized into three parts, including a non-integer pre-exponential power factor, dominant decaying terms on the exponent, and a modulator function, which is of minor importance. This result is a generalization of our previously obtained exact formula for the 1-body wave function. As in the 1-body case, here we show that for many-body ground wave functions that cannot achieve variable separation, our method is still far more efficient than the conventional basis expansion method both in representing the wave function and in calculating the energy. The exact factorization ansatz proposed in the present case study shall provide important insight into the general structure of many-body wave functions.
Recent studies have considered the possibility that the exact ground-state wavefunction from any Hamiltonian with two-particle interactions may be generated from a single finite two-body exponential transformation acting on an arbitrary Slater determinant [Piecuch et al., Phys. Rev. Lett. 90, 113001 (2003)]. Using the Campbell-Baker-Hausdorff relation, we show that it is difficult for the variational minimum of this trial wave function to satisfy the contracted Schr\"odinger equation which is a necessary and sufficient condition for the wave function to satisfy the Schr\"odinger equation. A counterexample is presented through the Lipkin quasispin model with 4--50 fermions. When the number of fermions exceeds four, the wave function from a finite two-body exponential transformation is shown to be inexact. If the trial wave function ansatz is extended to include products of finite two-body exponential transformations acting on an arbitrary Slater-determinant reference, then we show that the ansatz includes the exact ground-state wave function from any Hamiltonian with only two-particle interactions. Connections between the two-body exponential transformation of the wave function and recent research on two-body exponential similarity transformations of the Hamiltonian [S.R. White, J. Chem. Phys. 117, 7472 (2002)] are discussed.
The Schrödinger equation serves as a pivotal differential equation, capturing the fundamental duality of particles as both matter and waves. Upon solving it, we derive a particle's wave function within a given potential, delineating the confines within which the particle operates. This wave function enables us to ascertain the "most probable" values for parameters such as position, momentum, and energy within the specified potential. In this study, we focus on the double-well potential in two dimensions, resembling a bowl-like structure. Specifically tailored equations, termed Equation 1 and Equation 2, stem from the general Schrödinger equation when applied to the prescribed conditions. This research aims to utilize these equations to describe a particle’s behavior within the ground and the first excited states of the double well potential, employing two distinct methodologies. The first methodology involves leveraging the variational method on Equations 1 and 2 to obtain approximate solutions, providing crucial inputs for our subsequent approach. The second methodology serves as entails an analytical solution of Equations 1 and 2. A comparative analysis of the outcomes from both methods is anticipated to ensure the coherence and validity of the variational method. This project integrates key principles and proficiencies from both physics and mathematics disciplines, offering a platform for honing practical skills in applying these methodologies within a real-world context.
The minimum parameterization of the wave function is derived for the time-independent many-body problem. In this parameterization all electrons and electron pairs are indistinguishable which eliminates the exponential scaling with the number of particles plaguing all other correlation methods. An exponential scaling shown to stem from the expansion of the wave function in one-particle basis sets. It is demonstrated that using a geminal basis with a skew-symmetric orthogonal coefficient matrix, which fulfill a Lie algebra, the parameterization of the exact wave function becomes independent of the number of particles and only scale quadratic with the number of basis functions in the optimized basis and quartic in an arbitrary basis. The resulting antisymmetrized geminal power wave function is shown to fulfill the necessary and sufficient conditions for the exact wave function, treat all electrons and electron pairs as indistinguishable and be the most compact representation of the exact wave function possible. The number of parameters in the antisymmetrized geminal power wave function is exactly the same number of parameters as in the two-particle reduced density matrix thereby showing that these representations are equivalent just like the representations for a one-particle Hamiltonian. A detailed discussion of how the most compact wave function can be derived in general is also presented along with different geminal ansatze and sufficient closure relations. Due to the breaking of the scaling wall for the exact wave function it is expected that, if a general set of constraints can be found, even systems of biological relevance can be treated exactly in the future.
Coupled cluster (CC) theory is the “gold standard” of molecular electronic structure theory, yet the full arc of the subject—from its variational origins in Hartree–Fock theory, through the exponential ansatz and its diagrammatic machinery, to its modern reformulation as a system of polynomial equations on an algebraic variety, its multireference generalizations, and its emerging reincarnation as a quantum circuit—is rarely presented as a single connected narrative. This review builds that narrative completely from first principles and carries it through to today’s research frontier. Starting from the electronic Schrödinger equation, we derive Hartree–Fock theory, second quantization, and Wick’s theorem; construct the coupled cluster singles-and-doubles (CCSD) equations via the similarity-transformed Hamiltonian; and show how these same equations reappear, verbatim, as a truncated eigenvalue problem on a projective variety generalizing the Grassmannian, with a computable CC degree controlling the number of solutions. We then develop the perturbative triples correction CCSD(T), the linear-response equationof-motion (EOM-CC) framework for excited, ionized, and electron-attached states, the effective-Hamiltonian formalism underlying multireference coupled cluster theory and its notorious intruder-state problem, and the unitary coupled cluster (UCC) ansatz that restores the variational principle at the cost of an infinite operator expansion—tamed, on a quantum computer, by Trotterization and the variational quantum eigensolver. We close with a synthesis of what is rigorously known about the resulting nonlinear equations, a map of directions deliberately left out of scope, and an assessment of the open question of quantum advantage in this domain. Every claim is stated formally, with a full statement and proof or proof sketch, and cited to its origin; the computational cost of every method is compared explicitly throughout.