On van der Waals forces. Ioannis Anapolitanos

On van der Waals forces by Ioannis Anapolitanos A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2011 by Ioannis Anapolitanos

Abstract On van der Waals forces Ioannis Anapolitanos Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2011 The van der Waals forces, which are forces between neutral atoms and molecules, play an important role in physics (e.g. in phase transitions), chemistry (e.g. in chemical reactions) and biology (e.g. in determining properties of DNA). These forces are of quantum nature and it is long being conjectured and experimentally verified that they have universal behaviour at large separations: they are attractive and decay as the inverse sixth power of the pairwise distance between the atoms or molecules. In this thesis we prove the van der Waals law under the technical condition that ionization energies (energies of removing electrons) of atoms are larger than electron affinities (energies released when adding electrons). This condition is well justified experimentally as can be seen from the table, Atomic number Element Ionization energy (kcal/mol) Electron affinity (kcal/mol) 1 H 313.5 17.3 6 C 259.6 29 8 O 314.0 34 9 F 401.8 79.5 16 S 238.9 47 17 Cl 300.0 83.4 where we give ionization energies and electron affinities for a small sample of atoms, and is obvious from heuristic considerations (the attraction of an electron to a positive ion ii

is much stronger than to a neutral atom), however it is not proved so far rigorously. We verify this condition for systems of hydrogen atoms. With an informal definition of the cohesive energy W (y), y = (y 1,..., y M ) between M atoms as the difference between the lowest (ground state) energy, E(y), of the system of the atoms with their nuclei fixed at the positions y 1,..., y M and the sum, M j=1 E j, of lowest (ground state) energies of the non-interacting atoms, we show that for y i y j, i, j {1,..., M}, i j, large enough, W (y) = 1,M i<j σ ij y i y j + O( 1,M 1 6 y i y j ) 7 where σ ij are in principle computable positive constants depending on the nature of the atoms i and j. i<j iii

Acknowledgements I am grateful to my PhD thesis supervisor Professor Israel Michael Sigal for suggesting this problem and for his guidance and help throughout my work on the thesis. The thesis comes from a paper that is a joint work with him. I am also grateful to our wonderful secretary Mrs. Ida Bulat for her unconditional commitment and care to me and all my colleagues. I would also like to thank my PhD committee members Professors Almut Burchrd and Catherine Sulem for the advice that they gave me from time to time. In addition, I would like to thank Professors Alessandro Giuliani, Jim Colliander and Avy Soffer, and my colleague Artem Dudko for fruitful discussions. I also want to thank my family for all the support that they have shown to me during my PhD years. Finally I would like to thank all my friends who have in there own ways supported me and helped me. iv

Contents 1 Introduction 1 2 Preliminaries about many body systems 11 2.1 Decompositions................................ 11 2.2 Spherical symmetry of eigenfunctions of H A................ 13 3 The general approach (without statistics) 15 3.1 Feshbach map and existence of F P...................... 15 3.2 Orthogonal projection P and existence of F P................ 16 3.3 Estimate of P HP............................... 17 3.4 Rough bounds on E............................. 18 3.5 Estimate of U(E).............................. 22 3.6 Conclusion of the argument......................... 27 4 Technical results (no statistics) 28 4.1 Proof of (3.9) assuming condition (E).................... 28 4.2 Proof of (3.35)................................ 36 4.3 Proof of (3.49) and (3.50).......................... 46 4.4 Proof of (3.12)................................ 52 4.5 Proof of (3.45)................................ 56 4.6 Proof of Condition (E) for a system of several hydrogen atoms...... 59 v

5 Proof of Theorem 1.0.4 62 5.1 The general set-up in the case of statistics................. 62 5.2 Estimate of P HP............................... 64 5.3 Lower bound on H σ............................. 68 5.4 Rough bound on E σ (y)............................ 72 5.5 Estimate of U σ................................ 74 5.6 Completion of the proof of Theorem 1.0.4................. 90 5.7 Proof that Ran Q Ran P σ {0} and QP σ = P σ Q............. 91 6 Supplements 95 6.1 State space of fermions............................ 95 6.2 HVZ Theorem................................. 99 6.3 Born-Oppenheimer Approximation..................... 101 6.4 Non rigorous derivation of van der Waals law................ 107 vi

Chapter 1 Introduction Van der Waals forces between atoms play a fundamental role in quantum chemistry, physics and material sciences. A microscopic explanation of these forces was given by F. London soon after the discovery of quantum mechanics and was one of the early triumphs of the latter. This heuristic explanation showed that this force has a universal behavior at large distances - it decays as the inverse sixth power of the distance between atoms. So far this behaviour was confirmed in [LT] by proving - through a sophisticated test function construction - an upper bound. Our goal in this thesis is to prove this conjecture. Consider a system of M multielectron atoms which we call a molecule though we do not assume binding between atoms. In the units where = 1 the Hamiltonian of the system is H mol = M j=1 1 2m j yj + N i=1 ( 1 2m x i M j=1 e 2 Z j 1,N x i y j ) + i<j e 2 1,M x i x j + i<j Z i Z j e 2 y i y j. Here N is the total number of electrons, x i, y i R 3 denote the coordinates of the electrons and the nuclei, respectively, e and m are the electron charge and mass, ez j is the charge of the j-th nucleus and m j is the mass of the j-th nucleus. The operator H mol acts on the subspace H mol fermi of the space L2 (R 3(N+M) ), which accounts for the fact that the electrons are identical particles and are fermions therefore they obey the Fermi-Dirac statistics and possibly some of the nuclei are identical and obey either the Fermi-Dirac or Bose-Einstein 1

Chapter 1. Introduction 2 statistics (more details are discussed below). For M = 1, H mol is the Hamiltonian of the atom with the nucleus of charge ez = ez 1. To define the interaction energy of the system one fixes the positions of the nuclei and considers the Born-Oppenheimer Hamiltonian H N (y) = N i=1 ( 1 2m x i M j=1 e 2 Z j 1,N x i y j ) + i<j e 2 1,M x i x j + i<j Z i Z j e 2 y i y j, (1.1) where y = (y 1,..., y M ) is the collection of the nuclear co-ordinates, acting on the subspace H fermi of the space L 2 (R 3N ), which accounts for the Fermi-Dirac statistics of electrons. For M = 1 the operator above is independent of y and describes, as before, the atom with the nucleus of charge ez = ez 1. It arises a key technique in solving the eigenvalue problem for H mol, which is called the Born-Oppenheimer approximation. In this approximation one first holds the nuclei fixed which leads to the Born-Oppenheimer Hamiltonian. Then, the ground state energy E(y) (or the energy of an excited state) of H N (y) is considered as the potential energy of the nuclear motion, which leads to the Hamiltonian H nucl := M j=1 1 2m j yj + E(y). One expects that due to the fact that the ratio of the electron and nuclear mass being very small, the eigenvalues of H nucl give a good approximation to the eigenvalues of H mol. The Born-Oppenheimer Hamiltonian plays a key role in quantum chemistry. For example minima of its ground state energy E(y) (which depends naturally on the nuclear configurations y) determine shape of molecules. For rigorous results on the Born- Oppenheimer approximation see [CDS, KMSW, LeL, GS, Hag, PST]. The general information on the spectrum of the Hamiltonian (1.1) is given in the following Theorem which is a special case of the HVZ Theorem (see e.g. [HS, GS, CFKS]). Theorem 1.0.1. σ ess (H N (y)) = [Σ, ), Σ = inf σ(h N 1 (y)). This Theorem says that the essential (continuous) spectrum of H N (y) originates from the molecule shedding of an electron which moves freely at infinity and therefore whose

Chapter 1. Introduction 3 energy spectrum changes continuously. The next result shows that H N (y), as well as each atom, has a well-localized ground state (see e.g. [HS, GS, CFKS]): Theorem 1.0.2. The operator H N (y) has infinite number of eigenvalues, E j, below its essential spectrum, σ ess (H N (y)). Moreover, the corresponding eigenfunctions, Ψ j, are exponentially localized, Ψ j (x) Ce δ x, (1.2) for any δ < Σ E j. Here x = (x 1,..., x N ) and E j < Σ := inf σ ess (H N (y)) is the eigenvalue corresponding to Ψ j. The last Theorem shows that the atoms and Born-Oppenheimer molecules are stable in the sense that at sufficiently low energy they are well localized in the space. However, this Theorem says nothing about stability of true molecules. The likely source of instability of molecules is not shedding of an electron but breaking up into atoms or ions which in total have lower energy than the molecule. So far the only molecule known to be stable is the hydrogen molecule H 2 (see [BFGR]). The uniqueness of the ground state is a delicate issue. Without statistics the ground state of Schrödinger operators H is unique (non-degenerate). This follows from the positivity improving property of e βh, β > 0 and from Perron-Frobenious Theory (see for example [ReSIV]). For spaces with statistics the ground state energy is in general degenerate but its multiplicity is not known. Let E(y) be the ground state energy of the operator H N (y) and E m be the ground state energy of the m-th atom corresponding to the m-th nucleus of charge Z m. Let also E( ) = M E j, (1.3) j=1 i.e E( ) is the energy of the system with the atoms infinitely far from each other. Then the interaction (or cohesive) energy W (y) between the atoms in this system is defined as W (y) := E(y) E( ). (1.4)

Chapter 1. Introduction 4 It is expected, after van der Waals, that W (y) is a sum of pair interactions, W ij, which are attractive and decay at infinity as y i y j 6. More precisely, one expects that W (y) = 1,M i<j e 4 σ ij y i y j + O( 1,M e 4 ), (1.5) 6 y i y j 7 where σ ij are constants depending only on the pair of atoms i, j, provided that min{ y i y j : 1 i < j M} is large enough. First we consider van der Waals law on the subspace H A H Fermi of totally antisymmetric functions. Such functions have maximum possible spin. i<j Theorem 1.0.3 (van der Waals law; highest spin). Assume that the Hamiltonian H N (y) acts on the space H A of purely antisymmetric functions and assume Conditions (D) and (E) stated below. Then for large distances between atoms (1.5) holds for some constants σ ij > 0 depending on the nature of the atoms i, j. The Theorem above is a corollary of the Theorem 1.0.4 which is stated below. Using an intricate test function Lieb-Thiring [LT] have proven an upper bound for the interaction energy which is valid even for not too large separations. Now we define the physical state space, H fermi, of the Born-Oppenheimer molecule. Since electrons are identical particles and are fermions of spin 1, the state space of the 2 system of N electrons is the space N (L 2 (R 3 ) C 2 ) 1 of L 2 functions, Ψ(x 1, s 1,..., x N, s N ), of co-ordinates, x 1,..., x N, and spins, s 1,..., s N that are antisymmetric with respect to permutations of pairs (x i, s i ), (x j, s j ). The space H fermi is the subspace of L 2 (R 3N ) given by the projection of the space N 1 (L2 (R 3 ) C 2 ) onto the L 2 functions of the co-ordinates alone, H fermi := { χ, Ψ spin Ψ N (L 2 (R 3 ) C 2 ), χ : { 1 2, 1 2 }N C}, 1

Chapter 1. Introduction 5 where χ, Ψ spin := s 1,...,s N { 1 2, 1 2 } χ(s 1,..., s N )Ψ(x 1, s 1,..., x N, s N ). In terms of irreducible representations, T σ, of the group S N of permutations of N indices the space H fermi can be written as H fermi = σ H σ, (1.6) where σ runs over irreducible representations of the group S N corresponding to twocolumn Young diagrams and H σ is the subspace of L 2 (R 3N ) on which the unitary representation T : S N U(L 2 (R 3N )) (unitary operators on L 2 ) (T π Ψ)(x 1,..., x N ) = Ψ(x π 1 (1), x π 1 (2),..., x π 1 (N)) is multiple of the irreducible representation corresponding to σ (see Appendix 6.1 for definitions and details). We call irreducible representation labels σ the symmetry types. Next we need some notation. Let a be the decomposition of the molecule into neutral atoms, S(a) S N be the subgroup of S N consisting of the permutations that keep the clusters of a invariant and let H a be the sum of the corresponding atomic Hamiltonians (see the next Section for precise definitions). Fix an irreducible representation σ of the permutation S N. Then the restriction σ S(a) is not necessarily irreducible. Therefore, there exists a family of irreducible representations I σ such that T σ S N S(a) = α I σt α S(α). The representations α I σ are called induced representations and we write α σ. We write α σ if α σ and inf σ(ha α ) = min b,β inf σ(h β b ). We now denote the ground state energy of the system for the symmetry type σ by E σ (y) = inf σ(h H σ), and the corresponding energy of separated atoms as E σ ( ) = min a,α σ inf σ(h a H α).

Chapter 1. Introduction 6 The interaction energy for the symmetry type σ is defined as W σ (y) := E σ (y) E σ ( ). Now we formulate a Theorem which we need for the van der Waals law. Theorem 1.0.4 (van der Waals forces for a fixed symmetry type). Assume Conditions (D) and (E) below. Then for every symmetry type σ and every α σ, there exist positive constants σ σ,α ij (defined in (5.121)) such that W σ (y) = min α σ W σ,α (y), where W σ,α (y) := 1,M i<j σ σ,α ij y i y j + O( 1,M 1 6 y i y j ). 7 The ground state energy of H N (y) on H fermi is E(y) = min E σ (y) for σ corresponding to a Yonge tableaux of at most two columns. Let σ 0 be such that i<j E σ 0 (y) = min E σ (y) so that E(y) = E σ 0 (y). Then Theorem 1.0.4 for the specified σ 0 gives the interaction energy of the system. The Theorem describes the van der Waals force at a pairwise large separation between the atoms. The van der Waals energy is known to be repulsive (positive) for small distances due to overlapping between electron clouds of the atoms. In fact this also follows from the following simple argument: Since for any α > 0 there exists β > 0 such that e 2 Z m x n y m α xn + β, using the last relation and (1.1) we obtain that H N (y) C + 1,M i<j we obtain that e 2 Z i Z j y i y j, for some sufficiently large constant C independent of y. Therefore, E(y) as min{ y i y j : i, j = 1,..., M, i j} 0. (1.7) Often the interaction energy for M = 2 is modeled by the Lennard-Jones potential W LJ (y) = a y 1 y 2 12 b y 1 y 2 6 or by the Buckingham potential W B (y) = e c y 1 y 2 d y 1 y 2 6 where the constants a, b, c, d are determined experimentally.

Chapter 1. Introduction 7 The van der Waals law is expected to be true for the case of molecules that are not polarized (including the noble gases and methane). If the molecules have dipole moments then a third power law is expected to be true (see e.g [AT] Chapter 11). Now we will state Conditions (D) and (E) of Theorems 1.0.3 and 1.0.4: (D) For each α σ the ground state subspace of H α a consists only of one copy of the irreducible representation of type α. (E) For any two atoms i and j in our system, i j, and for m 1 < 0 < m 2 Z j and n > 0 with n min{ m 1, m 2 }, we have that E i,m1 +n + E j,m2 n < E i,m1 + E j,m2, where E m,n is the infimum of the spectrum of the Hamiltonian of the ion with a nucleus of charge ez m and Z m n electrons. One expects that for every symmetry type α, the ground state subspace consists of a single copy of the irreducible representation of the symmetry group, but proving this is an open problem. The meaning of condition (E) is that ionization energies of atoms are bigger than the electron affinities of atoms. The (first) ionization energy of an atom is the energy required to remove an electron from the atom. The electron affinity of an atom is the energy required to remove an electron from its -1 ion. The table below gives ionization energies and electron affinities of atoms. It is taken from [MA]. From the tables it is very clear that the ionization energies are always much larger than the electron affinities.

Chapter 1. Introduction 8 Atomic number Element Ionization energy (kcal/mol) Electron affinity (kcal/mol) 1 H 313.5 17.3 2 He 566.9-3 Li 124.3 (14) 4 Be 214.9-5 B 191.3 (7) 6 C 259.6 29 7 N 335.1-8 O 314.0 34 9 F 401.8 79.5 10 Ne 497.2-11 Na 118.5 (19) 12 Mg 176.3-13 Al 138.0 (12) 14 Si 187.9 (32) 15 P 254 (17) 16 S 238.9 47 17 Cl 300.0 83.4 18 Ar 363.4 (16) 19 K 100.1-20 Ca 140.9-21 Sc 151.3-22 Ti 158-23 V 155-24 Cr 156-25 Mn 171.4 -

Chapter 1. Introduction 9 (Values in parentheses are estimated by quantum-mechanical calculation and have not been verified experimentally.) For a system of several hydrogen atoms Condition (D) follows from the fact that S(a) is the trivial group. Also for a system of hydrogen atoms we provide a simple proof of Condition (E) in Appendix 4.6. Therefore, for systems of hydrogen atoms our result is unconditional. To simplify exposition we first treat the case without taking statistics into account and then modify our results and analysis for the case of an appropriate statistics. In this case the Condition (D) follows from the positivity improving property of e βh, β > 0 and from Perron-Frobenious Theory as discussed above. More specifically we first prove Theorem 1.0.5 (van der Waals forces without statistics). Consider all the Hamiltonians on the entire corresponding L 2 spaces. Assume Condition (E) stated above. Then for min{ y i y j : 1 i < j M} large enough (1.5) holds with the constants σ ij defined in (3.51). Condition (E) looks the same for both Theorems 1.0.4 and 1.0.5 but one has to bear in mind that the underlying spaces on which the Hamiltonians are defined, and therefore their ground state energies, depend on the underlying space and these spaces are different in the Theorems above. Again, proving this condition is an open problem. Note that when σ corresponds to a Young diagram with one row (completely symmetric representation) Theorem 1.0.4 gives Theorem 1.0.5. The thesis is organized as follows. In Chapter 2 we discuss preliminaries of quantum many body systems. In Chapter 3 we prove Theorem 1.0.5 modulo some technical relations which are proven in Chapter 4. In Chapter 5 we rework the proof of Theorem 1.0.5 to prove Theorem 1.0.4. Finally in Chapter 6 we discuss some well known results that are useful in the understanding of the Theory of van der Waals forces.

Chapter 1. Introduction 10 Notation. We collect here general notation used in this thesis. In the text C will denote a positive constant which might be different from one equation to the other. We will use the notation for inequalities that are true up to a constant that is uniform with respect to the variables involved in them. We will write A =. B if A B e CR, in an appropriate norm, where C is some positive constant and R = min{ y i y j : 1 i, j M, i j}. (1.8) We will write B(X) := {f : X X : f linear and bounded}, for any Banach space X. We also use the notation x = (1 + x 2 ) 1 2 and = N j=1 x j, = ( x1,..., xn ) with xj, xj the Laplacian and gradient acting on the coordinate x j, respectively (recall that x 1,..., x N are the electron coordinates). Finally,. will denote the L 2 (R l ) norm of a function or the B(L 2 (R l )) norm of an operator and the symbol O(δ) is understood in this norm. The dimension l will be in each case clear from the context. It will also be clear when. is operator norm or function norm.

Chapter 2 Preliminaries about many body systems 2.1 Decompositions Let a = {A 1,..., A M } be a partition of {1, 2,..., N} into disjoint subsets some of which might be empty. With the set A j we associate the j-th nucleus of charge ez j by assigning the electrons of coordinates in A j to be in the same atom/ion as the j-th nucleus. This gives a decomposition of the system. We denote the collection of all such decompositions by A and we will call A 1,..., A M clusters of the decomposition a. The set of all a A with A j = Z j for all j = 1,..., M will be denoted by A at. Its elements correspond to decompositions of our system to neutral atoms. If a = {A 1,..., A M }, b = {B 1,..., B M } A at, then there exists a permutation π S N /S(a) (here / denotes set theoretic difference)(where recall that N is the number of electrons and S(a) is the subgroup of S N consisting of the permutations that keep the clusters A 1,..., A M of a invariant) such that B j = {π 1 (i) i A j }, j {1,..., M}. (2.1) In this case we write b = πa. Various b s are related by permutations of the electrons and 11

Chapter 2. Preliminaries about many body systems 12 could be labeled as b = a π = πa, π S N /S(a). Note that we could use also more general K cluster decompositions and finer decompositions into > M subsystems, but we do not need this. For each decomposition a = {A 1,..., A M } A we define the Hamiltonian where, H Am := i A m ( xi H a = M H Am, (2.2) m=1 e2 Z m x i y m ) + i,j A m,i<j e 2 x i x j, (2.3) (so H Am is the Hamiltonian of the m-th atom or ion and H a is the sum of the Hamiltonians of the atoms or ions of the decomposition a) and the inter-atomic interaction I a := H(y) H a (in other words I a consists of all terms of interaction between the different atoms/ions in the decomposition a). We have that H(y) = H a + I a. (2.4) Let E a and Ψ a be the ground state energy and ground state of H a, respectively. Then H a Ψ a = E a Ψ a. If we ignore statistics of identical particles then E a = M m=1 E m where E m is the ground state energy of the m-th atom corresponding to the m-th nucleus. Also the ground state Ψ a of H a is given by Ψ a (x 1,..., x N ) = M φ Aj (x Aj ), (2.5) j=1 where for any j = 1,..., M, x Aj = (x i : i A j ) and φ Aj (x Aj ) = φ j (x Aj y j ), (2.6) with φ j the ground state of the j-th atom with the nucleus fixed at the origin and x Ak y j = (x i y j : i A j ). Throughout the text we will always assume that φ j = 1 for all j = 1,..., M. Note that by the definition of a s, E a = E b a, b A at. Standard estimates (see [HS]) give that there exists θ > 0 such that e θ x A j α φ j 1, α with 0 α 2, (2.7)

Chapter 2. Preliminaries about many body systems 13 where α is a multiindex with each index corresponding to differentiation in some body variable. For a A we define y a = (y a,1,..., y a,n ) where y a,i = y m for i A m. In other words y a,i is the coordinate of the nucleus that x i is assigned to. From (2.5), (2.6) and (2.7) we obtain that e θ x ya α Ψ a 1, α with 0 α 2. (2.8) 2.2 Spherical symmetry of eigenfunctions of H A Lemma 2.2.1. If Ψ is an eigenfunction of H A of an atom A with infinitely heavy nucleus, corresponding to a non degenerate eigenvalue then the one electron density of Ψ is spherically symmetric. In particular, the one electron density of the ground state of H A is spherically symmetric. Proof. Assume without loss of generality that the infinitely heavy nucleus is at the origin and the atom consists of k electrons. Then H A = k ( xj e2 1,k k x j ) + e 2 x i x j. j=1 i<j For any rotation R in R 3 we consider the transformation T R defined by T R Ψ(x 1,..., x k ) = Ψ(R 1 x 1,..., R 1 x k ). By the chain rule and the fact that columns of the matrix of R form an orthonormal basis of R 3 we obtain that the Laplacian in R 3 commutes with R. Since in addition Rx i Rx j = x i x j we obtain that H A commutes with T R or H A T R = T R H A. Since Ψ is eigenfunction of H A the last relation gives that T R Ψ is also an eigenfunction of H A corresponding to the same eigenvalue. Since the eigenvalue is non degenerate we

Chapter 2. Preliminaries about many body systems 14 obtain that T R Ψ = c(r)ψ, where c(r) is a complex valued function. Since T R is unitary we have that c(r) = 1 for any R and therefore, Ψ(x 1,..., x k ) 2 = Ψ(Rx 1,..., Rx k ) 2, for any rotation R. As a consequence, Ψ(x 1, x 2,..., x k ) 2 dx 2...dx k = Ψ(Rx 1, Rx 2..., Rx k ) 2 dx 2...dx k. Applying on the second integral the change of variable y j = Rx j, j = 2,..., k and taking into account that the Jacobian for this change of variable is 1 we obtain that Ψ(x 1, x 2,..., x k ) 2 dx 2...dx k = Ψ(Rx 1, y 2,..., y k ) 2 dy 2...dy k. or after renaming the variables Ψ(x 1, x 2,..., x k ) 2 dx 2...dx k = Ψ(Rx 1, x 2,..., x k ) 2 dx 2...dx k. This gives the spherical symmetry of the one electron density of any eigenfunction corresponding to a nondegenerate eigenvalue. From Perron Frobenius theory (see e.g [ReSIV]) it follows that the ground state energy of H A is a nondegenerate eigenvalue. Therefore, the one electron density of the ground state has to be spherically symmetric.

Chapter 3 The general approach (without statistics) In this Section we lay out the general set-up for our approach without taking the statistics into account. Our goal is to prove Theorem 1.0.5. In this Section we prove 1.0.5 under certain technical assumptions, which are then verified in later Sections for specific quantum systems. In what follows we omit the argument y and write E and H for E(y) and H(y), respectively. 3.1 Feshbach map and existence of F P Let P be an orthogonal projection and P = 1 P. Introduce the notation H = P HP. We will use the Feshbach-Schur method (see [BFS, GS]), which, as applied to the quantum Hamiltonian H, states that if Ran(P ) D(H) (domain of H) and HP < The operator (H λ) is invertible; then the Feshbach-Schur map F P (λ) = (P HP U(λ)) Ran P, (3.1) 15

Chapter 3. The general approach (without statistics) 16 where is well defined and U(λ) := P HP (H λ) 1 P HP, (3.2) λ eigenvalue of H λ eigenvalue of F P (λ). (3.3) (Here we do not display the y dependence of F P (λ)). Moreover, the eigenfunctions of H and F P (λ) corresponding to the eigenvalue λ are connected as follows. Define the family of operators Q(λ) as Q(λ) = P P (H λ) 1 P HP. (3.4) Then Hψ = λψ F P (λ)φ = λφ, where φ, ψ are related by the following equations φ = P ψ, ψ = Q(λ)φ. (3.5) In what follows we will explain how we are going to use the Feshbach map. The first step is to choose the orthogonal projection P. 3.2 Orthogonal projection P and existence of F P Now, we define P to be the orthogonal projection on span{ψ a : a A at }. We use this projection in the Feshbach-Schur method described above. According to (3.1), we have to compute P HP and U(λ). To compute P HP and U(λ), we use that the functions Ψ a are almost pairwise orthogonal for large R (where R is defined in (1.8)), i.e that Ψ a, Ψ b =. δ a,b, following from (2.8). This implies that the matrix (g ab ) := ( Ψ a, Ψ b ) =. 1. Hence, we can write P = Ψ a g ab Ψ b, (3.6) a,b A at

Chapter 3. The general approach (without statistics) 17 where (g ab ) is the inverse of the matrix (g ab ). Due to the fact that (g ab ). = 1, we have that this inverse satisfies As a consequence, we have (g ab ). = 1. (3.7) P =. Ψ a Ψ a. (3.8) a A at We will now prove existence of the Feshbach map. First of all we have that Ran(P ) Dom(H) since the range of P is spanned by (Ψ a ) a A at and by (2.8) each Ψ a is in H 2 (R 3N ). In addition by (3.6) we have that HP = a,b A at H Ψ a g ab Ψ b = and since each Ψ a is in H 2 (R 3N ) we have that HP <. a,b A at HΨ a g ab Ψ b Assume we can show that there is γ > 0, independent of y s.t. for R large enough H E( ) + 2γ, (3.9) where E( ) was defined in (1.3). We will prove this bound under Condition (E) in Section 4.1. Using the last relation we obtain that for all λ E( ) + γ (H λ) 1 1, (3.10) so H λ is invertible. Thus for all λ E( ) + γ, F P (λ) is well defined and satisfies the conditions of the Feshbach Schur method. Our goal now is to estimate the two terms on the r.h.s of (3.1). 3.3 Estimate of P HP Using (3.6) and (3.7) we obtain that P HP = Ψ a g ab Ψ b, HΨ c g cd Ψ d a,b,c,d A at

Chapter 3. The general approach (without statistics) 18. = a,c A at Ψ a Ψ a, HΨ c Ψ c. The last relation together with (3.8) and the fact that HΨ c = (H c + I c )Ψ c = (E( ) + I c )Ψ c, gives P HP. = E( )P + We will show in Appendix 4.4 that, for all a, b A at. Equations (3.11) and (3.12) imply that a,c A at Ψ a Ψ a, I c Ψ c Ψ c. (3.11) Ψ a, I b Ψ b. = 0, (3.12) P HP. = E( )P. (3.13) 3.4 Rough bounds on E Before proceeding to analysis of U(λ) we show that M 1=i<j 1 y i y j 6 E E( ) O(e CR ). Note that the left hand side gives a bound from below of the van der Waals interaction energy which is sharp in the order of magnitude. Due to (3.9) U(λ) is well-defined, for λ E( )+γ, and is positive definite, U(λ) 0. Next, using (3.6), (3.7), HΨ a = (H a +I a )Ψ a = (E( )+I a )Ψ a and that P E( )Ψ a = 0, we obtain P HP =. P I a Ψ a Ψ a. (3.14) a A at (Note that due to (2.8) the operators on the r.h.s and l.h.s. are bounded.) Similarly we obtain that P HP. = Ψ a I a Ψ a P. (3.15) a A at The relations (3.2), (3.14) and (3.15) give that U(λ). = a,b A at Ψ a U ab (λ) Ψ b, (3.16)

Chapter 3. The general approach (without statistics) 19 where Due to (3.6), (3.7) and (3.12) we obtain that U ab (λ) := Ψ a, I a P (H λ) 1 P I b Ψ b. (3.17) P I a Ψ a. = Ia Ψ a, P I b Ψ b. = Ib Ψ b, (3.18) which together with (3.17), imply that for all λ E( ) + γ we have U ab (λ). = I a Ψ a, (H λ) 1 I b Ψ b. (3.19) We show in Appendix 4.5 that I a Ψ a L 2 M 1=i<j 1 y i y j 3, (3.20) which together with (3.2), (3.9), (3.16), (3.19), (3.10) and the arithmetic mean-geometric mean inequality implies that for λ E( ) + γ, U(λ) 0, m λ U(λ) M 1=i<j 1, m 0, (3.21) y i y j 6 uniformly in λ E( )+γ (the inequality U(λ) 0 follows merely from (3.2) and (3.9)). Next, we want to show Lemma 3.4.1. The ground state energy E of H satisfies the following inequalities: M 1=i<j 1 y i y j 6 E E( ) e CR. (3.22) Proof. The right hand side of (3.22) follows immediately since using the relations (2.4), H a Ψ a = E( )Ψ a and (3.12) we obtain that Ψ a, HΨ a = Ψ a, H a Ψ a + Ψ a, I a Ψ a = E( ) + Ψ a, I a Ψ a E( ) + e CR. We will now show the left hand side of (3.22). Let λ E( )+γ, where γ > 0 is the same as in (3.9). Then by (3.2) and (3.9), U(λ) is well-defined and is positive definite, U(λ) 0. Therefore, the finite rank operator

Chapter 3. The general approach (without statistics) 20 F P (λ, y) := F P (λ) defined in (3.1) is well defined (recall that H and therefore F P (λ) depends on y). Let ν(λ, y) be the lowest eigenvalue of F P (λ, y). We consider the fixed point problem λ = ν(λ, y) on the interval (, E( ) + γ]. We will show below that for R large enough the fixed point problem has a unique solution λ 0 (y). By (3.1), (3.2), (3.13) and (3.21), we have that ν(λ, y) E( ) M 1=i<j and λ ν(λ, y) M 1=i<j 1 y i y j 6 1 y i y j uniformly in λ E( ) + γ. Hence, for R sufficiently 6 large, ν is a contraction and maps (, E( ) + γ) to [E( ) γ, E( ) + γ]. Therefore, it is enough to consider the fixed point problem on the interval [E( ) γ, E( ) + γ]. Since ν is a contraction on this interval and maps it to itself the fixed point problem has a unique solution λ 0 (y) in [E( ) γ, E( ) + γ]. From (3.1), (3.2), (3.13), (3.21) we obtain that F P (λ 0 (y)) E( )P M 1=i<j and since λ 0 (y) is eigenvalue of F P (λ 0 (y)) we obtain that M 1=i<j 1 y i y j 6 P 1 y i y j 6 λ 0(y) E( ). (3.23) Also by isospectrality of the Feshbach map with the Hamiltonian H we obtain that λ 0 (y) is eigenvalue of H. We will now show that E = λ 0 (y), (3.24) i.e that λ 0 (y) is the ground state of H which together with (3.23) will imply the left inequality in (3.22). Indeed, since λ 0 (y) is eigenvalue of H and E is the lowest eigenvalue of H we have that E λ 0 (y). (3.25)

Chapter 3. The general approach (without statistics) 21 So it is enough to show that E λ 0 (y). (3.26) First we show that for fixed y F P (λ) is a decreasing function of λ on (, E( ) + γ] or in other words that λ 1 < λ 2 = F P (λ 1 ) F P (λ 2 ), λ 1, λ 2 (, E( ) + γ]. (3.27) Indeed, by (3.1) and (3.2) it suffices to show that (H λi) 1 is increasing function of λ on (, E( ) + γ]. Indeed, for any λ 1, λ 2 (, E( ) + γ] with λ 1 < λ 2 the operator (H λi) 1 is defined bounded and differentiable on [λ 1, λ 2 ] and therefore by the Fundamental Theorem of Calculus (H λ 2 ) 1 = (H λ 1 ) 1 + λ2 λ 1 (H λ) 2 dx and since (H λ) 2 = ((H λ) 1 ) 2 0 we obtain that (H λi) 1 is an increasing function of λ on (, E( ) + γ] and we can conclude that (3.27) holds. From (3.25) and (3.27) we obtain that F P (λ 0 (y)) F P (E). (3.28) Since we have that F P (λ 0 (y)) λ 0 (y) (because λ 0 (y) is the lowest eigenvalue of F P (λ 0 (y))) the relation (3.28) gives that λ 0 (y) F P (E). (3.29) The last relation together with the fact that E is eigenvalue of F P (E) gives (3.26). The relations (3.25) and (3.26) give (3.24). The relations (3.23) and (3.24) give the left inequality of (3.22). This completes the proof of (3.22). The estimate (3.9) together with (3.22) gives that for R sufficiently large H E c > 0, (H E) 1 1. (3.30) Consequently, due to its definition (3.17), it follows that the matrix (U ab (E)) is welldefined and non-negative, (U ab (E)) 0, and uniformly bounded.

Chapter 3. The general approach (without statistics) 22 3.5 Estimate of U(E) Let y ij = y i y j. Our goal is to show that where, f(y) = U(E). = f(y)p, (3.31) 1,N i<j e 4 σ ij y ij (1 + O( 1 )), (3.32) 6 R where σ ij > 0 is a constant depending only on the nature of the atoms and is determined below (see relation (3.51)). Recall that x = (x 1,..., x N ) is the collection of the electron coordinates. Let φ(y) = φ R (y) be a C 2 monotonically increasing function of x with uniformly bounded derivatives of first and second order (in both x, R) and satisfying, x, if x R 1 ϕ(x) = R, if x R + 1. (3.33) Recall y a = (y a,1,..., y a,n ) where y a,i = y m for i A m and a decomposition a = {A 1,..., A M } A at. In other words y a,i denotes the coordinate of the nucleus that the electron with coordinate x i is assigned to by the decomposition a. For any operator K let K δ := e δϕ(x ya) Ke δϕ(x ya), Kδ := (K ) δ, (3.34) where recall that z = (1 + z 2 ) 1 2, and δ > 0. We show in Section 4.2 that (Hδ E) 1 1. (3.35) We use this result to show that we have that U ab (E) =. 0 a, b A at with a b. (3.36) Indeed, using (3.19) and the relation e δϕ(x ya) (H E) 1 e δϕ(x ya) = (Hδ E) 1, (3.37)

Chapter 3. The general approach (without statistics) 23 we have that U ab (E) =. e δϕ(x ya) I a Ψ a, (Hδ E) 1 e δϕ(x ya) e δϕ(x yb) e δϕ(x yb) I b Ψ b, (3.38) where δ > 0. Due to (2.8) we have for all δ < θ that e δϕ(x ya) I a Ψ a, e δϕ(x yb) I b Ψ b 1. (3.39) In addition, since by the construction of ϕ (see (3.33)) we have that ϕ(x y k ) min{ x y k, R 1} we obtain that ϕ(x y a ) + ϕ(x y b ) min{r 1, x y a + x y b }. Using the triangle inequality x y a + x y b y a y b R (since a b), we obtain that ϕ(x y a ) + ϕ(x y b ) R 1, and therefore we have that e δϕ(x ya) e δϕ(x yb) L e δr. (3.40) Using the relations (3.35), (3.38), (3.39) and (3.40) we arrive at (3.36). Now we show that U aa (E) = U bb (E), a, b A at. (3.41) Indeed, for any a, b A at there exists a permutation π with b = πa (this equality was defined precisely in (2.1)). For any Φ L 2 (R 3N ) and for any permutation π S N /S(a), where recall that N is the number of electrons and S(a) is the subgroup of permutation group S N consisting of permutations that keep the clusters A 1,..., A M of the decomposition a invariant, we define T π Φ(x 1,..., x N ) = Φ(x π 1 (1),..., x π 1 (N)). Then, we have that HT π = T π H, and P T π = T π P π S N. (3.42) From the last relation it follows that if Φ eigenfunction of H then T π Φ is also eigenfunction of H with the same eigenvalue.

Chapter 3. The general approach (without statistics) 24 By the fact that b = πa we obtain that T π I a Ψ a = I b Ψ b. Using (3.17), and the last relation we obtain that U bb (E) = T π I a Ψ a, P (H E) 1 P T π I a Ψ a. The last relation together with (3.42) implies that U bb (E) = T π I a Ψ a, T π P (H E) 1 P I a Ψ a. Finally the last relation together with the unitarity of T π and (3.17) implies the relation (3.41). The relations (3.8), (3.16), (3.36) and (3.41) and imply that U(E). = U aa (E)P. (3.43) Thus it suffices to estimate U aa (E) for a fixed a. For the latter, it suffices by (3.19) to estimate I a Ψ a, (H E) 1 I a Ψ a. For a = {A 1,..., A M } we have with I Al A m = i A l,j A m I a = M 1=l<m e2 x i x j i A l e 2 Z m x i y m I Al A m, (3.44) j A m e2 Z l y l x j + e2 Z l Z m y l y m, consists of the interaction terms of the atoms A i, A j (recall that ez j is the charge of the j-th nucleus). We introduce the variables z qr = x q y r, q A r, r = 1,..., M, y r1 r 2 = y r1 y r2, r 1, r 2 {1,..., M}. We will show in Appendix 4.5 that I Ai A j Ψ a = e2 y ij f 1 3 A i A j (z, ŷ ij )Ψ a + O( ), (3.45) y ij 4

Chapter 3. The general approach (without statistics) 25 (the remainder here is in the sense of the L 2 norm) where ŷ ij = y ij y ij and f Ai A j (z, ŷ ij ) = k A i,l A j (z ki z lj 3(z ki ŷ ij )(z lj ŷ ij )). (3.46) We insert (3.45) into (3.19) with b = a and use (3.30) to obtain U aa (E) = α,β e 4 σ a α,β y α 3 y β 3 + O( α,β e 4 ), (3.47) y α 3 y β 4 where α, β run in pairs of nuclei in the decomposition a and σ a αβ = f α Ψ a, (H E) 1 f β Ψ a. (3.48) We prove in Section 4.3 that and that, for α = (ij), where σα,β a 1, if α β, (3.49) R σα,α a = σ ij + O( 1 ), (3.50) R σ ij = f ij φ i φ j, (P ij (H Ai + H Aj )P ij E i E j ) 1 f ij φ i φ j (3.51) and we wrote f ij for f Ai,A j and recall that H Am is the Hamiltonian of the m-th atom (see relation (3.52) below), E m its ground state energy and P ij is the projection onto the ground state of H Ai + H Aj. The relations (3.43), (3.47), (3.49) and (3.50) imply (3.31) modulo the fact that σ ij are positive and independent of y which we prove below. Lemma 3.5.1. The quantity σ ij defined in (3.51) (which is a priori function of y) does not depend on y and is positive. Proof. First we note that the only dependence of σ ij on ŷ ij := y i y j y i y j appears on f ij so we will write f ŷij ij and σŷij ij. For any rotation R in R3 we define T R acting on L 2 (R 3( A i + A j ) ) (recall that A k is equal to the number of electrons of the k-th atom), or T R g(z 1i, z 2i,..., z Ai i, z 1j,..., z Aj j)

Chapter 3. The general approach (without statistics) 26 = g(r 1 z 1i, R 1 z 2i,..., R 1 z Ai i, R 1 z 1j,..., R 1 z Aj j) where z kl = x k y l. In the z variables z kl = x k y l by (2.3) we obtain that H Am = k A m ( ) zkm e2 Z m + e 2 z km z km z lm. (3.52) k,l A m,k<l Therefore T R commutes with H Ai + H Aj. So since φ i φ j is a ground state of H Ai + H Aj obtain that T R φ i φ j is also eigenfunction of H Ai + H Aj. Therefore, since the ground state is non degenerate we obtain that T R φ i φ j = c(r)φ i φ j where c(r) = 1. This implies that T R φ i φ j T R φ i φ j = φ i φ j φ i φ j and as a consequence we T R Pij T 1 R = P ij. Using the last relation and the fact that H Ai + H Aj commutes with T R we obtain that T R (P ij (H Ai + H Aj )P ij E i E j ) 1 T R 1 = (P ij (H Ai + H Aj )P ij E i E j ) 1. The last relation together with (3.51) and the fact that T R is unitary imply that σŷij ij = T R 1(f ŷij ij φ iφ j ), (Pij (H Ai + H Aj )Pij E i E j ) 1 T R 1(f ŷij ij φ iφ j ). (3.53) On the other hand using (3.46) and the fact that R is unitary we obtain that T 1 R f ŷij ij R = f 1 y ij ij. (3.54) But using that T 1 R φ iφ j = c(r 1 )φ i φ j with c(r 1 ) = 1 together with (3.53) and (3.54) we obtain that T R 1(f ŷij ij φ iφ j ) = c(r 1 )f R 1 y ij ij φ i φ j. The last relation together with (3.53) implies that σŷij ij R 1 y ij ij = f φ i φ j, (P ij (H Ai + H Aj )P R 1 y ij ij ij E i E j ) 1 (f φ i φ j ) = σ Ry ij ij

Chapter 3. The general approach (without statistics) 27 which implies that σ ij is independent of y ij. Therefore, it suffices to prove that σ ij > 0. By (3.51) and the fact that P ij (H Ai + H Aj )P ij E i E j is a positive operator we obtain that σ ij 0. Assume now that σ ij = 0 then since (P ij (H Ai + H Aj )P ij E i E j ) 1 0 and by (3.51) and the relation inf σ(a) = inf Ψ, AΨ Ψ X where X is a Hilbert space A a self-adjoint operator on X and σ(a) the spectrum of A, we have that f ij φ i φ j is an eigenfunction of (P ij (H Ai + H Aj )P ij E i E j ) 1 with eigenvalue zero. This implies that f ij φ i φ j = (P ij (H Ai + H Aj )P ij E i E j )(P ij (H Ai + H Aj )P ij E i E j ) 1 f ij φ i φ j = 0, which is a contradiction. 3.6 Conclusion of the argument From (3.1), (3.13) and (3.31) we have that F P (E). = (E( ) f(y))p. (3.55) Since the ground state energy E is an eigenvalue of H we conclude from (3.3), (3.32) and (3.55) that E E( ) = i,j e 4 σ ij y i y j 6 + O( i<j e 4 ). (3.56) y i y j 7 Therefore, the interaction energy W (y) = E E( ), is given by (1.5). Since by the definition, E( ) := M m=1 E m, we have proven the following result: Theorem 3.6.1. If the relations (3.12), (3.9), (3.35), (3.45), (3.49) and (3.50) hold, then so does the van der Waals law (Theorem 1.0.5).

Chapter 4 Technical results (no statistics) 4.1 Proof of (3.9) assuming condition (E) Recall the system we consider consists of M atoms with Z 1,..., Z M electrons, respectively. Let γ 1 = min{e j E j : j = 1,..., M}, (4.1) with E j, E j the ground state energy and the first excited state energy of the atom j (corresponding to the j-th nucleus), respectively. Recall that E j,nj denotes the ground state energy of the ion of the atom j with charge n j e and E( ) was defined in (1.3). Let γ 2 = min{(e 1,n1 +... + E M,nM ) E( )}, (4.2) where the minimum is taken among all n 1,..., n M Z, with n 1 +... + n M = 0, n i Z i, n 1 + n 2 +... + n M 0. Now we show that Condition (E) implies that for all integers n 1, n 2,..., n M with n 1 +... + n M = 0, n j Z j, j = 1,..., M, and n 1 + n 2 +... + n M 0, we have that E 1,n1 +... + E M,nM > E 1 +... + E M, (4.3) where, recall, E m,n is the infimum of the spectrum of the Hamiltonian of the ion with a nucleus of charge ez m and Z m n electrons. 28

Chapter 4. Technical results (no statistics) 29 We prove (4.3) by induction in subcollections {1, 2,..., k} of the collection of the atoms {1,..., M}, i.e. in k. For k = 2 the result follows immediately. We assume that (4.3) holds for k 1. We will show it holds for k. Indeed, let n 1,..., n k be numbers satisfying the assumptions of condition (4.3) for the atoms {1,..., k}. Assume without loss of generality that n 1 n k and n 1 n k < 0. Then by condition (E) we have that E 1,n1 + E k,nk > E 1,n1 n k +E k. Therefore, E 1,n1 +...+E k 1,nk 1 +E k,nk > E 1,n1 n k +E 2,n2 +...+E k 1,nk 1 +E k which together with the induction hypothesis implies (4.3) for k. Due to (1.3) and (4.3) we have that γ 2 > 0. Also let γ 0 := min{γ 1, γ 2 } > 0. (4.4) The relation (3.9) will follow immediately from the following proposition: Proposition 4.1.1. H E( ) + γ 0 δ(r), where δ(r) 0 as R. In particular (3.9) holds for R large enough. Proof. Recall that N is the total number of electrons in the system of atoms we consider. Recall also that A is the set of all decompositions, which correspond to all partitions a = {A 1,..., A M } of {1,..., N} with A j 0 for all j. Thus, A includes decompositions of the system with clusters that are possibly ions. We will split the configuration space R 3N into domains. For each a = {A 1,..., A M } A we define Ω β a = {(x 1,..., x N ) : x i y j βr, j = 1,..., M, i / A j }. (4.5) Following [S] we will now construct a partition of unity (J a ) a A satisfying the following properties for all a A: 0 J a 1, supp J a Ω 1 5 a, J a L 1 R, J a L 1 R 2, (4.6) and the relation Ja 2 = 1. (4.7) a A

Chapter 4. Technical results (no statistics) 30 We consider the functions F a = χ 3 Ω 10 φ where φ : R 3N R is a Cc function supported a in B 1 ( 0) (ball of radius 1 centered at the origin) with φ 0 and φ = 1 and χ 10 10 R 3N A denotes the characteristic function of the set A. Then F a C and F a 0. Furthermore using the triangle inequality and the fact that φ is supported in B 1 ( 0) we obtain that 10 supp(f a ) Ω 1 5 a and that F a Ω 2 5a = 1. The last relation together with the fact that a A Ω 2 5 a = R 3N gives that a A F a 1. Therefore, if we can define J a = F a a A F, a 2 then J a is a partition of unity satisfying all the desired properties. Remark 1. By taking φ to be symmetric with respect to particle coordinates we can make J a commute with all elements of S(a) where S(a) is the group of permutations that keeps the clusters of a invariant. Now we use the IMS localization formula (see for example [CFKS]) H = a A ( Ja HJ a J a 2). (4.8) This together with (4.6) gives that H a A J a HJ a C R. (4.9) The last relation together with (2.4), and I a M N e 2 j=1 i=1,i/ A C on supp J j x i y j R a implies that P HP a A P J a H a J a P C R. (4.10) We will now estimate each of the terms on the right hand side of (4.10). Lemma 4.1.2. We have that, a A, P J a H a J a P (E( ) + γ 0 )P J 2 ap C R, (4.11) where γ 0 was defined in (4.4).

Chapter 4. Technical results (no statistics) 31 We will consider the following two cases Case (1): a A/A at in other words some clusters of the decomposition a include ions. In this case we define n j := Z j A j. Then the Hamiltonian H Aj defined in (2.3) satisfies the relation H Aj E j,nj, since recall that by definition E j,nj denotes the infimum of the spectrum of H Aj. The last relation together with (4.2), (4.4) and (2.2) gives that H a E 1,n1 +... + E M,nM (E( ) + γ 0 ) and as a consequence P J a H a J a P (E( ) + γ 0 )P J 2 ap. (4.12) Case (2): a A at in other words a is a decomposition of the system to M (neutral) atoms. Now, let P a = Ψ a Ψ a, (4.13) where recall that Ψ a is the ground state of H a. We claim P J a H a J a P. = P a J a H a J a P a. (4.14) Pa J a H a J a Pa J a Pa H a Pa J a C R. (4.15) J a Pa H a Pa J a (E( ) + γ 0 )J a Pa J a. (4.16) J a Pa J a P JaP 2 C R. (4.17) The lemma follows immediately from the claims, so it suffices to prove all the claims.

Chapter 4. Technical results (no statistics) 32 Proof of (4.14). We have that P J a H a J a P Pa J a H a J a Pa = (P Pa )J a H a J a P + Pa J a H a J a (P Pa ). So it suffices to show that (P Pa )J a H a, H a J a (P Pa ) =. 0. (4.18) We will prove that (P Pa )J a H a =. 0 and the relation H a J a (P Pa ) =. 0 can be proven similarly. Indeed, we have that (P Pa )J a H a = (P a P )J a H a. Therefore, using (3.6), (4.13) and the self-adjointness of J a, H a, we obtain that (P Pa )J a H a = Ψ a H a J a Ψ a Ψ d g db H a J a Ψ b. (4.19) d,b A at We will now prove that H a J a Ψ b =. 0, a, b A at, a b. (4.20) Indeed, by the Laplacian boundedness of the Coulomb potential we have that H a J a Ψ b J a Ψ b + (J a Ψ b ) J a Ψ b + J a Ψ b + J a Ψ b + J a Ψ b. (4.21) Therefore, it suffices to show that J a Ψ b, J a Ψ b, J a Ψ b, J a Ψ b =. 0. (4.22) We will show that J a Ψ b =. 0 and the other relations can be prove similarly. Multiplying by e θ x y b e θ x y b and using the inequality fg L 2 f L g L 2 we obtain that J a Ψ b J a e θ x y b L e θ x y b Ψ b,

Chapter 4. Technical results (no statistics) 33 where θ is the same as in (2.8). The last relation together with (2.8) implies that J a Ψ b J a e θ x y b L. (4.23) Since J a is supported on Ω 1 5 a, by (4.5) and (4.6) we obtain that 0 J a e θ x y b J a e θr 5 e θr 5. (4.24) The relations (4.23) and (4.24) imply that J a Ψ b. = 0. Proceeding similarly we can complete the proof of (4.22) which together with (4.1) implies (4.20). The relations (3.7), (4.19) and (4.20) imply that (P P a )J a H a. = Ψa H a J a Ψ a Ψ a g aa H a J a Ψ a. = 0. This completes the proof of (4.14). Proof of (4.15). Using the relation P a J a = J a P a + [P a, J a ] = J a P a [P a, J a ] we obtain that P a J a H a J a P a = J a P a H a P a J a + [J a, P a ]H a J a P a + J a P a H a [P a, J a ]. Therefore, to prove (4.15) it is enough to show that [J a, P a ]H a, H a [P a, J a ] 1 R. (4.25) We will show that [J a, P a ]H a 1 R, (4.26) and the relation H a [P a, J a ] 1 R can be shown similarly. Since P a = Ψ a Ψ a and H a is self-adjoint we have that [J a, P a ]H a = ( J a Ψ a Ψ a Ψ a J a Ψ a )H a

Chapter 4. Technical results (no statistics) 34 = J a Ψ a H a Ψ a Ψ a H a (J a Ψ a ). Using now that H a Ψ a = E( )Ψ a together with the Leibnitz rule and the relation (4.6) we obtain that [J a, P a ]H a = E( )( J a Ψ a Ψ a Ψ a J a Ψ a ) + O( 1 R ). Subtracting and adding E( )Ψ a Ψ a we obtain that [J a, P a ]H a = E( )( (J a 1)Ψ a Ψ a Ψ a (J a 1)Ψ a ) + O( 1 R ). Therefore, it suffices to show that (1 J a )Ψ a. = 0. (4.27) Indeed, using (2.8) and multiplying by e θ x ya e θ x ya we obtain that (1 J a )Ψ a = (1 J a )e θ x ya L e θ x ya Ψ a (1 J a )e θ x ya L. Therefore, it is enough to show that (1 J a )e θ x ya L. = 0. (4.28) Indeed, since 0 (1 J a )e θ x ya (1 + J a )(1 J a )e θ x ya we obtain that (1 J a )e θ x ya L (1 J 2 a)e θ x ya L, which if we use the relation (4.7) gives that (1 J a )e θ x ya L Jb 2 e θ x ya L. (4.29) b A,b a Now since by (4.6) we have that x y a R on the support of J 5 b for all b a we obtain that 0 Jb 2 e θ x ya Jb 2 e θ R 5 e θ R 5, (4.30) b A,b a b A,b a where in the last step we used (4.7). The relations (4.29) and (4.30) give (4.28).

Chapter 4. Technical results (no statistics) 35 Proof of (4.16). If a = {A 1,..., A M } then by (2.5) we can write P a = M j=1 P A j where P Aj is the orthogonal projection onto the function φ Aj defined in (2.6). Since for all j = 1,..., M we have that P Aj commutes with H a and P A j P Aj = P Aj P A j = 0 and P a = M j=1 (P A j M i=j+1 P A i ), we obtain that P a H a P a = M M M (PA j ( P Ai )H a PA j ( P Ai )). j=1 i=j+1 i=j+1 The last relation together with the relation (2.2) for the decomposition a gives that P a H a P a = M M M M (( P Ai )PA j H Al PA j ( P Ai )). l=1 j=1 i=j+1 i=j+1 Since all the projections on the right hand side commute and H Al E l (recall that E l is the ground state energy of the atom corresponding to the nucleus l) and P A l H Al P A l (E l + γ 1 )P A l (by the definition of γ 1 in (4.1) and the fact that P A l removes the ground state energy from the spectrum of H Al ) we obtain that P a H a P a M M ( P Ai )PA j (E( ) + γ 1 ) = Pa (E( ) + γ 1 ). j=1 i=j+1 The last relation together with (4.4) completes the proof of (4.16). Proof of (4.17): Using that (P a ) 2 = P a and that [P a, J a ] = [P a, J a ] we obtain that J a P a J a = J a (P a ) 2 J a = P a J 2 ap a + [P a, J a ]P a J a P a J a [P a, J a ] The last relation together [P a, J a ] 1, which can be proven if we proceed as in the R proof of (4.26), give that Proceeding as in the proof of (4.18) we can obtain that J a P a J a P a J 2 ap a C R. (4.31) (P P a )J a, J a (P P a ). = 0. (4.32) The relations (4.31) and (4.32) imply (4.17). Equations (4.7), (4.10) and (4.11) give that H (E( ) + γ 0 )P C R.