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METHODS FOR THE DIRECT CALCULATION OF REDUCED DENSITY MATRICES by . John Peter Simons A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Chemistry) at the University of Wisconsin 1970 METHODS FOR 'SHE DIRECT CALCUTATION OF RF,DUCED >k DENSTTY MATRICES c John Peter Simons Under the supervision of Associate Professor John E. Harriman ABSTRACT This thesis contains the results of four separate efforts to overcome some of the mathematical difficulties involved in formulating atomic and molecular quantum mechanics in terms of reduced density matrices. First the problem of wnstructing approximately N-representable density matrices which can be used in variational calculations is studied in detail. The effects of approximate N-gepresentability on calculated properties are also analyzed. Secondly use is made of field theoretical Green's functions to directly determine the sectond-order density matrix of an N-fermion + NSF Graduate Fellow * This work was supported in part by the National Aeronautics and Space Administration Grant NGL 50-002-001, system, This method is applied to the ground state of the helium atom as a test case. Thirdly a new technique for calculating, in a self-consistent fashion, the first-and second-order density matrices of atoms and molecules is put forth. This scheme makes use of a generalized random-phase approximation to obtain equations for various spin components of the density matrices. The results of applying this new method to helium, lithium, and beryllium are presented. Finallyg the first-and second-order density matrices of symmetry-projected single determinants are derived and analyzed. In memory of my friend David Guzaliak iii ACKNOWLEDGMENTS I would like to express my sincere appreciation to my colleagues in the Theoretical Chemistry Institute for their encouragement, help, and friendship. I will always remember fondly these happy and productive years at the University of Wisconsin. I owe special debts of gratitude to the following people: Professor John E. Harriman for his wise and patient guidance of my graduate research and for his helpful criticism of my ideas; Professor Joseph 0. Hirschfelder for providing such pleasant and stimulating surroundings and for his willingness to help the younger scientists in the institute; My wife Peg for her critical and accurate proofreading of my manuscripts and for not complaining about sharing her eccentric husband with science, his other love; The staff of the Theoretical Chemistry Institute for their help and friendship iv TABLE OF CONTENTS Dedication is Acknowledgments iii INTRODUCTION 1 Chapter One. CONSTRUCTION OF APPROXINATELY N-REPRESEN!UB%E DENSITY MATRICES 8 I. Introduction 9 II. Development of Formalism 12 III.Consequenees of Approximate N-Representability 26 IV. Methods of Calculation and Effects of Truncation 31 V. Discussion of Results 38 Appendix 1. Spin orbital expansion of the spin gemfnals 42 A Appendix 2. The Relationship between and T. 44 Notes and References 46 Chapter Two. DIRECT CALCULATION OF SECOND ORDER DENSITY MATRICES USING GUEN'S FUNCTIONS 49 I. Introduction 50 II. Second-Order Density Matrix and 'Two-Particle Green's Function 50 V III. Time-dependent Perturbation Expansion of iYY> and Diagrams 56 IV. Evaluation of Two-Particle Green's Function 61 Q. ERPA in a basis 80 Notes and References 85 Chapter Three. DIRECTION CALCULATION OF FIRST-AND SECOND-ORDER DENSITY MATRICES USING A GENE'KALIZED WDOM-PHASE APPROXIMATION 86 I. Introduction 87 II The Generalized Random-Phase Approximation 89 IT1 0 Density Matrices in the Occupation Number Representation 99 IV* Use of the GRPA 102 Q. First-Order Density Matrix 107 VI. Self-consistent Determination of 108 ro,rz,y0 , and yz VI1 . Application to Helium, Lithium, and Beryllium 110 VIII.Error Bounds 117 IX. Conclusions 125 Notes and References 128 Chapter Four. FIRST-AND SECOND-ORDER DENSITY MATRICES OF SYMMETRY-PROJECTED SINGLE DETERMINANT WAVE FUNCTIONS 132 vi I. Introduction 133 11. Derivation of the Density Matrices for Point-Group Projection 133 140 111 B Example of Finite Point-Groups IV . The Axial Rotation Group 144 V. Example for Axial Rotation Croup 144 VI. The Totally Symmetrfc Components of &?Y 146 VI1* Totally Symmetric Components of Pirst- Order Density Matrices Before and After Projeetion 148 VIII. Discussion of Results 15 1 References 15 3 1 INTRODUCTION For a quantum mechanical system composed of N identical, pairwise-interacting particles, the Hamiltonian operator can be written as follows: me one-particle operator f(i) operates only on the space-spin variables of the ith particle, which are represented by the index 1. Similarly the two-particle operator g( i,j) operates on the space-spin variables of both the ithand jth particles. The system wave function '1E/ (1,2,3,. ..N) is the solueion of the time-dependent Schrgdinger equation with the initial condition ai: t=t,.$ (39 If the Hamiltonian does not depend explicitly Qn time, the time- 11 dependent Schrodinger equation allows the separation of variables 2 I1 which leads to the time-independent Schrodinger equation H Y= EY (5) The solution of Eqn. (5) describes a stationary state of the system whose energy is equal to the eigenvalue E. The expectation value of any observable Q for a system described by the wave function is given as 4 where Q is the quantum mechanical operator corresponding to the observable Q, and d"t: implies integration over the space-spin variables of all N particles. If the particleg which comprise the system are fermions (bosons) then the wave function (1,2*3,. .N) must be antisymmetric (symmetric) under permutation of the space-spin variables of any two particles: This fact allows the expectation value expressed is Eqn. (6) to be h written in a more suggestive form. For example, if the operator Q is a one-particle operator 3 then Eqn. (6) can be written as follows: Eqn. (9) follows directly from Eqn. (6) by repeated application of the following identity: The first equality in Eqn. (10) is obtained by relabeling dummy integration variables; the second equality is a consequence of Eqn, (7). Arguments similar to those employed above can ba used to rewrite the expectation value of a two-particle operator in the form given below; where (:) is the binomial coefficient. It: is clear from Eqns. (9) and (11) that knowledge of the M-particLe 4 wave function is not necessary for the calculation of expectation values of one-and two-particle operators. All of the necessary information Is contained in tho first-and second-order reduced density matrices defined as follows: and Higher order density matrices are defined in an analogous fashion. In terms of these reduced quantities, the expectation values given in Eqns. (9) and (11) can be written as and For an N-particle system the wave function depends on 3N continuous variables and N spin varhbleg whereas ?he second-m-dew 5 density matrix r is a function of only twelve continuous variables and 4 spin variables,independent of the number of particles. Moreover the second-order density matrix contains a11 of the information needed to calculate the properties of systems composed of pairwise-interacting particles. These facts, together with the knowledge that accurate wave functions are extremely difficult to obtain for any but the simpJest systemsg lead naturally to investigating the possibility of determining directly the first-and second-order density matrices. By using the definitions of the reduced density matrices, e.g, Eqns. (12) and (El), and the fact that the wave function obeys the I1 Schrodinger equation, a system of coupled iategro-differential equattons involving the density matrices can be derived. In statistical mechanic6 this hierarchy of equations is known as the BBGKY (Bogolubov, Born, Green, Kirkwood, Yvon) equations. The main difficulty with such alp approach is that the equations are coupled; the equation which should be used to determine the first-order density matrix contains the secpnd- order density matrix. Likewise the equation for the second-order density matrix is coupled to the third-order density matrixs and so QV. There are many decoupling procedures which can be used to approximate the solutions of the first few equations in the hierarchy, but they are not based on sound theoretical foundations. Another approach to the direct determination of first-anad seappdr arder density matrices would be to use trial density matrices in 8 variational calculation. The difficulty with this approach is that the trial density matrices must be restricted to the class of functions which are derivable from an antisymmetric or symmetric wave functi~na~ in Eqns. (12) or (13). Such density matrices are said to be N-represpf- able. The problem of restricting trial density matrices to be N-reprp- sentable is very difficult and has yet to be solved. Presented in this thesis are the results of four separate efforts to develop techniques allowing the N-particle wave function to be by- passed in favor of reduced density matrices. En the first chapter the problem of constructing approximately N-representable density matrices which can be usad in variational calculations is studied in detail. The i. effects of approximate N-representability on calculated properties are also analyeed. In chapter two use is made of field-theoretical Green's functions to directly determine the second-order density matriees of N-fermion systems. As a test case the method is applied to the ground state of the helium atom. Chapter three contains the derivation of a new technique for calculating, in a self-consistent fashion, the first-and second-order density matrices of atoms and molecules. This new method makes use og a generalized random-phase approximation to obtain equations for various spin components of the density matrices. These equations are solved $+ by an iterative procedure.

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