Since H is a four-body Hamiltonian, the non-separability of V ceases the possibility of obtaining any exact solution for Equation (19), and we are obliged to choose between different approximations. The most mathematically reliable one is the conventional Rayleigh-Ritz’ variational method [48] . It is the first variational method of bound state type which has been employed in the treatment of the ground states of atomic, molecular and nuclear systems. It was also extended by Hylleraas-Undheim (see [48]) to the treatment of the excited states of any quantum mechanical system.

The method can be displayed as follows: consider the non-relativistic time independent quantum mechanical system defined in the preceding section, The Schrödinger’s equation is equivalent in form to the conventional eigenvalue problem:

H | ψ 〉 = E | ψ 〉 , (22)

or

( H − E ) | ψ 〉 = 0 , (23)

where E and H are the total energy and Hamiltonian, respectively, of a quantum mechanical system described by the vector | ψ 〉 . The Schrödinger constraint can be stated according to Equation (23) as follows: a true physical system or process described by the observable ( H − E ) is well expressed, microscopically, by the expansion space | ψ 〉 if and only if Schrödinger vector ( H − E ) | ψ 〉 defines a null space. Also, Equation (23) fulfills the variational principle:

∂ 〈 ψ | H − E | ψ 〉 = 0 (24)

and possesses the eigenvalues:

E k = 〈 ψ k | H | ψ k 〉 / 〈 ψ k | ψ k 〉 (25)

Equations (23), (24) and (25) imply a one-to-one correspondence between the E k ’s and | ψ k 〉 ’s once the degeneracy has been removed. Consequently, one can order the E k ’s such that

E k ≤ E k + 1 ,   for     k = 1 , 2 , ⋯ , (26)

where E 1 is the first (lowest) eigenvalue.

Now, the verification of Schrödinger’s constraint requires the exact knowledge of the terms E, H and | ψ 〉 . In practice, however, | ψ 〉 is always unknown and the parameter E is not given for the boundary value problems. For this, there was a necessity of using approximate methods to get a solution for physical problems.

In Rayleigh-Ritz variational method a trial expansion space | ψ t ( n ) 〉 is selected which defines a hypothetical physical system such that

ψ t ( n ) = ∑ k n a k   | ψ t k 〉 (27)

where n is the dimension of | ψ t k ( n ) 〉 , and

〈 ψ t k ( n ) | ψ t k ′ ( n ) 〉 = δ k k ′ ,   for   k ,   k ′ = 1 , 2 , ⋯ , n (28)

where δ k k ′ is the Kroneker-delta. Equation (28) will then reduce to

∂ 〈 ψ t ( n ) | H − E | ψ t ( n ) 〉 = 0 (29)

All | ψ t k ( n ) 〉 ’s are, due to Rayleigh-Ritz variational method, generated from one basis set of vectors { | χ i 〉 } ⊂ D H where D H is the H-domain, i.e.,

| ψ t k ( n ) 〉 = ∑ i = 1 n c i k | χ i 〉 (30)

Consequently, Equation (29) can be written for each k as the system of secular equations:

∑ j = 1 n c j k [ 〈 χ i | H | χ j 〉 − E n k 〈 χ i | χ j 〉 ] = 0 ,   i = 1 , 2 , ⋯ , n (31)

which is meaningful if and only if the determinant Δ n k satisfies the relation

Δ n k = det ( H i j − E n k S i j ) = 0 , (32)

where

H i j = 〈 χ i | H | χ j 〉 , (33a)

and

S i j = 〈 χ i | χ j 〉 (33b)

The eigenvalues obtained by (31) are ordered such that:

E n 1 ≤ E n 2 ≤ ⋯ ≤ E n n , (34)

is satisfied.

Rayleigh and Ritz (see e.g. [34] [35]) proved the important relation between E n 1 and the first exact energy level of the system E 1 , namely that

E 1 ≤ E n 1 ,   for   n > 0 , (35a)

i.e., for any choice of the components | ψ t k ( n ) 〉 , the first variational energy is an upper bound to the exact one. Hylleraas and Undheim (see e.g. [34]), moreover, have shown that if the condition (34) is fulfilled, we then get

E k ≤ E n   k ,   for   k = 1 , 2 , ⋯ , n . (35b)

McDonald (see also [48] and references therein), on the other hand, demonstrated that, if the trial expansion space is enlarged by exactly one component such that

〈 ψ t k ( n + 1 ) | ψ t k ′ ( n + 1 ) 〉 = δ k k ′ ,   for   k , k ′ = 1 , 2 , ⋯ , n + 1 , (36)

then the following successive relations are always valid:

E n + 1   1 ≤ E n   1 ≤ E n + 1   2 ≤ ⋯ ≤ E n + 1   n ≤ E n   n ≤ E n + 1   n + 1 , (37a)

E n + 1   k ≤   E n + 1   k ′   for   k ≤ k ′ , (37b)

E n + 1   k + 1 ≤   E n   k ′   for   k > k ′ , (37c)

E k ≤ E n + 1   k ≤ E n   k ≤   E n − 1   k ≤ ⋯ ≤ E 1   k . (37d)

Equations (37) imply that the existence of any negative E n k ensures the existence of corresponding bound state of the four-body system, and for this reason we are led to say that Rayleigh-Ritz’ variational method provides an approximate solution of (22).

The four-body theorem states that “If σ is the mass ratio characteristic to any particle-antiparticle pair, then the existence of two molecules with σ = 0 and σ = 1 is a sufficient condition for the existence of all molecules with σ lying between 0 and 1”.

To prove this theorem, let us go back to Equations (6) and (8) and define the following quantities: E 1 ( σ ) is the ground-state energy of the whole system, and W ( σ ) is the binding energy with which the four-body system is stable against any possible dissociation. Thus, it is obvious that the molecule is bound if and only if W ( σ ) satisfies the relation:

W ( σ ) = E 1 ( σ ) − E t h r < 0 . (38)

Now, from (16) we have

∂ H ∂ σ = 1 ( 1 + σ ) 2 ( ∇ 1 2 + ∇ a 2 − ∇ 2 2 − ∇ b 2 ) (39)

and

∂ 2 H ∂ σ 2 = − 2 1 + σ ∂ H ∂ σ (40)

Remember that H is the total Hamiltonian of the system without any restriction, but the one regarding the definition of the unit energy and unit length. On the other hand, Equation (8a) yields

∂ ω ∂ σ = ∂ ε ∂ σ     and     ∂ 2 ω ∂ σ 2 = ∂ 2 ε ∂ σ 2 . (41a)

While Equations (6) and (8b) imply

∂ W ∂ σ = ∂ E ∂ σ     and     ∂ 2 W ∂ σ 2 = ∂ 2 E ∂ σ 2 . (41b)

Now, let us restrict ourselves to the ground state of the four-body system and assume that E 1 ( σ ) and | ψ 1 〉 are, respectively, the corresponding exact ground state energy and wavefunction. Therefore, they satisfy the relations:

H | ψ 1 〉 = E 1 | ψ 1 〉     or     ( H − E 1 ) | ψ 1 〉 = 0 , (42)

which imply

∂ ( H − E 1 ) ∂ σ | ψ 1 〉 = − ( H − E 1 ) | ∂ ψ 1 ∂ σ 〉 (43)

and

〈 ψ 1 | ∂ ( H − E 1 ) ∂ σ | ψ 1 〉 = − 〈 ψ 1 | ( H − E 1 ) | ∂ ψ 1 ∂ σ 〉 (44)

Since H is Hermitian, the right-hand side of (44) is zero, and we have;

∂ E 1 ∂ σ = 〈 ψ 1 | ∂ H ∂ σ | ψ 1 〉 = 1 ( 1 + σ ) 2 〈 ψ 1 | ∇ 1 2 + ∇ a 2 − ∇ 2 2 − ∇ b 2 | ψ 1 〉 (45)

Defining the expectation values for the kinetic energy operators T a and T b as

〈 T a 〉 = 〈 ψ 1 | − 1 1 + σ ( ∇ 1 2 + ∇ a 2 ) | ψ 1 〉 〈 T b 〉 = 〈 ψ 1 | − 1 1 + σ ( ∇ 2 2 + ∇ b 2 ) | ψ 1 〉 . (46)

Equations (45) can then be written as:

∂ E 1 ∂ σ = 1 1 + σ ( 〈 T b 〉 σ − 〈 T a 〉 ) . (47)

Since | ψ 1 〉 is symmetric with respect to the coordinates of the two particles a and b, we expect that 〈 T a 〉 equals 〈 T b 〉 at σ = 1 , so that

∂ E 1 ∂ σ | σ = 1 = 0 . (48)

Using (41b), we get

∂ W 1 ∂ σ | σ = 1 = 0 . (49)

Now, operate with 〈 ∂ ψ 1 ∂ σ | on (43) from the left, we get:

〈 ∂ ψ 1 ∂ σ | ∂ ( H − E 1 ) ∂ σ | ψ 1 〉 = − 〈 ∂ ψ 1 ∂ σ | ( H − E 1 ) | ∂ ψ 1 ∂ σ 〉 (50)

The variational theory demands that (50)

〈 ∂ ψ 1 ∂ σ | ( H − E 1 ) | ∂ ψ 1 ∂ σ 〉 ≥ 0 . (51)

Therefore (50) leads to

− 〈 ∂ ψ 1 ∂ σ | ∂ ( H − E 1 ) ∂ σ | ψ 1 〉 ≥ 0 . (52)

On the other hand, Equation (43) provides us with

∂ ∂ σ [ ∂ ( H − E 1 ) ∂ σ | ψ 1 〉 ] = − ∂ ∂ σ [ ∂ ( H − E 1 ) ∂ σ | ∂ ψ 1 ∂ σ 〉 ] . (53)

∴   ∂ 2 ( H − E 1 ) ∂ σ 2 | ψ 1 〉 + ∂ ( H − E 1 ) ∂ σ | ∂ ψ 1 ∂ σ 〉 = − ∂ ( H − E 1 ) ∂ σ | ∂ ψ 1 ∂ σ 〉 − ( H − E 1 ) | ∂ 2 ψ 1 ∂ σ 2 〉

or

∂ 2 ( H − E 1 ) ∂ σ 2 | ψ 1 〉 + 2 ∂ ( H − E 1 ) ∂ σ | ∂ ψ 1 ∂ σ 〉 + ( H − E 1 ) | ∂ 2 ψ 1 ∂ σ 2 〉 = 0 (54)

Operating on this equation by 〈 ψ 1 | from the left, we obtain

〈 ψ 1 | ∂ 2 ( H − E 1 ) ∂ σ 2 | ψ 1 〉 + 2 〈 ψ 1 | ∂ ( H − E 1 ) ∂ σ | ∂ ψ 1 ∂ σ 〉 + 〈 ψ 1 | ( H − E 1 ) | ∂ 2 ψ 1 ∂ σ 2 〉 = 0. (55)

Again, the Hermeticity of H yields

〈 ψ 1 | ( H − E 1 ) | ∂ 2 ψ 1 ∂ σ 2 〉 = 0 , (56)

so that

〈 ψ 1 | ∂ 2 ( H − E 1 ) ∂ σ 2 | ψ 1 〉 = − 2 〈 ψ 1 | ∂ ( H − E 1 ) ∂ σ | ∂ ψ 1 ∂ σ 〉

using (52) we get

〈 ψ 1 | ∂ 2 ( H − E 1 ) ∂ σ 2 | ψ 1 〉 ≥ 0

so that

∂ 2 E 1 ∂ σ 2 ≤ 〈 ψ 1 | ∂ 2 H ∂ σ 2 | ψ 1 〉 , (57)

substituting from (45) into (57), we find

∂ 2 E 1 ∂ σ 2 ≤ − 2 1 + σ 〈 ψ 1 | ∂ H ∂ σ | ψ 1 〉 , (58)

using (45)

∂ 2 E 1 ∂ σ 2 ≤ − 2 1 + σ ∂ E 1 ∂ σ , (59)

and using (41a, 41b)

∂ 2 W 1 ∂ σ 2 ≤ − 2 1 + σ ∂ W 1 ∂ σ , (60)

Now, Equation (47) states that ∂ E 1 ∂ σ , and consequently ∂ E 1 ∂ σ , is a continuous function of σ in the interval (0, 1). This continuity together with the boundary condition (49) and the inequality (60) demand

∂ W 1 ∂ σ ≥ 0   for   all     σ ∈ ( 0 , 1 ) , (61)

that is to say W 1 ( σ ) is a monotonic function of σ in this interval. Also (60) and (61) lead to the inequality:

∂ 2 W 1 ∂ σ 2 ≤ 0   for   all   σ ∈ ( 0 , 1 ) . (62)

Thus W 1 ( σ ) is a concave function with the characteristics

W 1 ( 0 ) ≤ W 1 ( σ ) ≤ W 1 ( 1 )   for   all   σ ∈ ( 0 , 1 ) (63)

and it is bound within the triangle ( 0 , W 1 ( 0 ) ) , ( 0 , W 1 ( 1 ) ) , ( 1 , W 1 ( 1 ) ) , see Figure 2.

The inequality (63) states that: if W 1 ( 0 ) and W 1 ( 1 ) are negative, then W 1 ( σ ) will be negative at all σ ∈ ( 0 , 1 ) . That is to say if the four-body system has a ground state at σ = 0 , which corresponds in case I with Z = 1 , to the adiabatic treatment of H₂ molecule, and at σ = 1 , corresponds, at the same condition, to the positronium molecule Ps₂ that is a bound state composed of

two electrons and two positrons, it has a ground state at all σ ∈ ( 0 , 1 ) . In fact, it has been shown [28] [29] [30] that, starting from the inequality (56), the existence of a four-body system with σ = 1 is a sufficient condition for the existence of any molecule with 0 ≤ σ ≤ 1 , so that the existence of a positronium molecule for example implies the existence of molecule composed of a hydrogen and antihydrogen. Further, more the following points have been concluded:

1) If the four-body system defined by the Hamiltonian (16), exists at σ = 0 and σ = 1 , then all molecules with σ lying between 0 and ¥ should exist.

2) If W k = E k − 2 E a b k , where E k a b is the k^th state of the pair m a − , m b + , the following two conclusions are true:

a) If the k^th state of m 1 − m 2 + m 3 − m 4 + exists at σ = 0 and σ = 1 , the k-th state of m 1 − m 2 + m 3 − m 4 + exists at all 0 ≤ σ ≤ 1 .

b) W k ( 0 ) ≤ W k ( σ ) ≤ W k ( 1 ) (64)

The above theorem and conclusions suggest that systems like e − π + π − e + , e − μ + μ − e + , μ − π + π − μ + , ⋯ etc. may also exist as molecular structures.

On the other hand, the preceding theorem has been extended [30] to the cases at which M is replaced by nM, where n > 1 , which predicts the existence of systems like antihydrogen-deuterium ( H ¯ D ) and antihydrogen-tritium ( H ¯ T ) as bound states. Figure 3 shows a general schematic diagram for systems with M is replaced by nM where n > 1 .

For these systems, the Hamiltonian (1) is replaced by:

H = − ℏ 2 2 m e   ( ∇ 1 2 + ∇ a 2 ) − ℏ 2 2 M p ∇ 2 2 − ℏ 2 2 n M p ∇ b 2     + Z 2 e 2 ( 1 r 12 + 1 r a b − 1 r 1 a − 1 r 2 a − 1 r 1 b − 1 r 2 b ) , (65)

Also, Equation (16) takes the following form

H = { − 1 1 + σ [ ∇ 1 2 + ∇ a 2 + σ ( ∇ 2 2 + 1 n ∇ b 2 ) ] } T     + { 2 r 12 + 2 r a b − 2 r 1 a − 2 r 2 a − 2 r 1 b − 2 r 2 b } V , (66)

where n = 2 for H ¯ D and n = 3 for H ¯ T molecules.

The binding energy W 1 ( σ ) for such molecules will be given by:

W 1 ( σ ) = E 1 ( σ ) − E t h r − Δ (67)

where

Δ = ( n − 1 ) σ 2   ( n σ + 1 ) (68)

It is clear that Δ is always positive definite, so if W 1 ( σ ) is negative for n = 1 , it is necessarily negative for all n > 1 . This means that if the four-body molecule m 1 − m 2 + m 3 − m 4 + exists at n = 1 for a given σ , it exists at all values of n for the same σ . This in turn means that the existence of H ¯ H molecule implies the existence of H ¯ D and H ¯ T molecules. In particular, the following conclusions are valid:

1) If W k = E k − E k a 2 − E k 1 b where E k a 2 and E k 1 b are both the k-th excited states of the clusters a-2 and 1-b, respectively, and E k is the corresponding singlet excited state of the four-body system, the following statements are true:

a) If the k-th excited state of 12ab exists at σ = 1 , n = 1 , then the k-th excited state of 12ab exists also at 0 ≤ σ ≤ 1 for all n ≥ 1 .

b) W k ( 0 ) ≤ W k ( σ ) ≤ W k ( 1 ) .

The proof of a) follows from the definition of W k and the generalization of the theorem for k > 1 , while b) is a result of a) and is a generalization of (64).

This theory predicts the upper bound energies E k v to the real bound state energies E k by replacing the coordinates r i j by α r i j where α is a variational parameter. Since the kinetic energy operator is second order in ( ∂ ∂ r i j ) while the potential energy operator is first order of ( 1 r i j ), we may define the virial Hamiltonian as

H v = 1 α 2 T + 1 α V (69)

and since the virial energy E k is

E k v = 〈 ψ t k ( n ) | H | ψ t k ( n ) 〉 〈 ψ t k ( n ) | ψ t k ( n ) 〉 ,

we have

E k v = 〈 ψ t k ( n ) | 1 α 2 T + 1 α V | ψ t k ( n ) 〉 〈 ψ t k ( n ) | ψ t k ( n ) 〉 = 1 α 2 T e x + 1 α V e x , (70)

where,

T e x = 〈 ψ t k ( n ) | T | ψ t k ( n ) 〉 〈 ψ t k ( n ) | ψ t k ( n ) 〉     and     V e x = 〈 ψ t k ( n ) | V | ψ t k ( n ) 〉 〈 ψ t k ( n ) | ψ t k ( n ) 〉 (71)

Now, the upper bound states are obtained by minimization with respect to α. Doing so, Equation (70) gives:

∂ E k v ∂ α = − 2 α 3 T e x − 1 α 2 V e x = 0 , (72)

and

α = − 2 T e x V e x . (73)

Substituting from (74) into (71), we obtain

E k v = − V e x 2 4 T e x 〈 ψ t k ( n ) | ψ t k ( n ) 〉 = − 〈 ψ t k ( n ) | V | ψ t k ( n ) 〉 2 4 〈 ψ t k ( n ) | T | ψ t k ( n ) 〉 〈 ψ t k ( n ) | ψ t k ( n ) 〉 , (74)

this is known as the virial energy.

Let us consider Figure 1 and recall the Hamiltonian (16)

H = { − 1 1 + σ [ ∇ 1 2 + ∇ a 2 + σ ( ∇ 2 2 + ∇ b 2 ) ] } T     + { 2 r 12 + 2 r a b − 2 r 1 a − 2 r 2 a − 2 r 1 b − 2 r 2 b } V ,

which can be written as

H = − 1 1 + σ [ T 1 + T a + σ ( T 2 + T b ) ] + V (75)

The kinetic energy operator contains terms of the form

T i = − ∇ i 2 ;     i = 1 , a , 2 , b (76)

In terms of the coordinate system given in Figure 1, the operator in (76) can be written as

∇ i 2 = ∑ j ≠ , i , j < k ( ∂ 2 ∂ r i j 2 + 2 r i j ∂ ∂ r i j ) + 2 ∑ j ≠ i ∑ k ≠ i cos θ i j , i k ∂ 2 ∂ r i j ∂ r i k (77)

and

cos θ i j , i k = r i j 2 + r i k 2 − r j k 2 2 r i j r i k (78)

Our system must be described in a more suitable system of coordinates; this is because the components of the spherical polar coordinates r i j are not orthogonal to each other and so highly dependent. Besides, the new system must take in consideration the correlation relation between different particles in the molecule. We notice that the interaction relation between different particles in our system is not one-to-one particle interaction; instead, it is one-to-two.

Let us replace the relative coordinates with a Hylleraas’ coordinate system consisting of confocal ellipses and hyperbolas defined as

s i = ( r i a + r i b ) / r a b ;   i = 1 , 2 (79a)

t i = ( r i a − r i b ) / r a b ;   i = 1 , 2 (79b)

s a = ( r 1 a + r 2 a ) / r 12 ,   s b = ( r 1 b + r 2 b ) / r 12 , (79c)

t a = ( r 1 a − r 2 a ) / r 12 ,   t b = ( r 1 b − r 2 b ) / r 12 , (79d)

u = r 12 / r a b ,   v = r a b . (79e)

As it is evident s i , s a and s b are constants on ellipses the distance between their two foci are r a b for s i and r 12 for s a and s b , while t i , t a and t b are constants on hyperbolas again the distances between their two foci are r a b for t i and r 12 for t a and t b . It is clear that s goes from 1 to infinity, t goes from −1 to 1, and v goes from 0 to infinity. In addition to these variables there are also the angles of rotation Φ ’s about the axis joining the two foci.

The point now is to write the Hamiltonian of the system in terms of these coordinates. The partial derivatives with respect to r i a and r i b can be expressed in terms of s i and t i as follows:

∂ ∂ r 1 a = ∂ s 1 ∂ r 1 a ∂ ∂ s 1 + ∂ t 1 ∂ r 1 a ∂ ∂ t 1 (80)

then

∂ ∂ r 1 a = 1 v ( ∂ ∂ s 1 + ∂ ∂ t 1 ) ,       ∂ 2 ∂ r 1 a 2 = 1 v 2 ( ∂ 2 ∂ s 1 2 + 2 ∂ 2 ∂ s 1 ∂ t 1 + ∂ 2 ∂ t 1 2 ) , (81a)

and so on. Finally, we have

∂ ∂ r 12 = 1 v ∂ ∂ u ,   ∂ 2 ∂ r 12 2 = 1 v 2 ∂ 2 ∂ u 2 (81b)

We also have

cos θ i j , i k = r i j 2 + r i k 2 − r j k 2 2 r i j r i k ,     i = 1 , 2 = v 2 4 [ ( s i + t i ) 2 + ( s i − t i ) 2 ] − v 2 2 v 2 4 ( s i + t i ) ( s i − t i ) ,

∴   cos θ i a , i b = s i 2 + t i 2 − 2 s i 2 − t i 2 . (82)

Now, the kinetic energy operator (for example) T₁ can be written as

T 1 = − 4 v 2 1 s 1 2 − t 1 2 [ ( s 1 2 − 1 ) ∂ 2 ∂ s 1 2 + ( 1 − t 1 2 ) ∂ 2 ∂ t 1 2 + 2 s 1 ∂ ∂ s 1 − 2 t 1 ∂ ∂ t 1 ]     − 1 v 2 [ ∂ ∂ u + 2 u + 2 ( cos θ 12 , 1 a + cos θ 12 , 1 b ) ∂ ∂ s 1     + 2 ( cos θ 12 , 1 a − cos θ 12 , 1 b ) ∂ ∂ t 1 ] ∂ ∂ u , (83)

Similar forms can be derived for T₂, T_a and T_b.

On the other hand, the potential energy operator which is given by:

V = 2 ( 1 r 12 + 1 r a b − 1 r 1 a − 1 r 2 a − 1 r 1 b − 1 r 2 b ) ,

will be modified to

V = 2 v [ ( 1 u + 1 ) − 4 s 1 s 1 2 − t 1 2 − 4 s 2 s 2 2 − t 2 2 ] (84)

Two forms for the volume element d τ could be distinguished (see Equation (79)):

1) v is variable:

d τ = v 8 64 ( s 1 2 − t 1 2 ) ( s 2 2 − t 2 2 ) d s 1 d s 2 d t 1 d t 2 d v sin θ v d θ v d φ v d Φ 1 d Φ 2 (85)

1 ≤ s i ≤ ∞ ,   − 1 ≤ t i ≤ 1 ,   0 ≤ v ≤ ∞ , (86)

0 ≤ θ v ≤ π ,   0 ≤ φ v ≤ 2 π ,   0 ≤ Φ i ≤ 2 π . (87)

2) r_ab is Constant: In this case, the volume element is reduced to

d τ = v 6 64 ( s 1 2 − t 1 2 ) ( s 2 2 − t 2 2 ) d s 1 d s 2 d t 1 d t 2 d Φ 1 d Φ 2 (88)

The most suitable form of the trial wavefunctions for describing the four-body system is the one developed from Hilbert space components depending on Hylleraas’ coordinates.

| χ j 〉 = s 1 m j s 2 n j e − α j ( s 1 + s 2 ) t 1 k j t 2 l j cosh [ β j ( t 1 − t 2 ) ] u p j v q j e − γ j v . (89)

The total wavefunction of the system is then defined as the superposition of these components

| Ψ k 〉 = ∑ j = 0 ∞ c k j | χ j 〉 , (90)

where k stands for the state of the system; k = 1 identifies the ground-state, k = 2 is the first excited state, etc. In practical calculations the number of superpositions is subjected to the convergence of the calculated energies and, therefore, limited to a certain value n.

Applying now the kinetic energy operators given at Equations (83) to the j-th component χ j of the wavefunction, Equation (89), we obtain the following relations:

T 1 χ j = 1 v 2 ( s 1 2 − t 1 2 ) { [ − 4 m j ( m j − 1 ) + 4 k j ( k j − 1 ) − 4 ( p j + 2 ) ( m j − k j )         + 4 ( α j 2 − β j 2 ) ] + [ 4 m j ( m j − 1 ) s 1 2 ] − [ 4 k j ( k j − 1 ) t 1 2 ] − 8 m j ( α j ) s 1     − 8 k j ( β j ) t 1 tanh [ β j ( t 1 − t 2 ) ] + [ 4 α j ( 2 m j + 2 + p j ) ] s 1

  + [ 4 β j ( 2 k j + 2 + p j ) ] t 1 tanh [ β j ( t 1 − t 2 ) ] − ( 4 α j 2 ) s 1 2   + ( 4 β j 2 ) t 1 2 + ( p j α j ) s 1 3 u 2 + ( p j β j ) t 1 3 u 2 tanh [ β j ( t 1 − t 2 ) ]   − [ p j ( m j + p j + k j + 1 ) s 1 2 u 2 ] + [ p j ( m j + p j + k j + 1 ) t 1 2 u 2 ]

  − ( p j α j ) s 1 t 1 2 u 2 − ( p j β j ) t 1 s 1 2 u 2 tanh [ β j ( t 1 − t 2 ) ] + [ p j ( m j − k j ) s 2 2 u 2 ]   + [ p j ( m j − k j ) t 2 2 u 2 ] − ( p j α j ) s 1 s 2 2 u 2 − ( p j α j ) s 1 t 2 2 u 2   − ( p j β j ) t 1 s 2 2 u 2 tanh [ β j ( t 1 − t 2 ) ] − ( p j β j ) t 1 t 2 2 u 2 tanh [ β j ( t 1 − t 2 ) ]   + ( 2 p j β j ) s 1 s 2 t 2 u 2 tanh [ β j ( t 1 − t 2 ) ] + ( 2 p j k j ) s 1 s 2 t 2 t 1 u 2   + ( 2 p j α j ) t 1 s 2 t 2 u 2 − ( 2 p j m j ) t 1 s 2 t 2 s 1 u 2 } χ j , (91a)

T 2 χ j = 1 v 2 ( s 2 2 − t 2 2 ) { [ − 4 n j ( n j − 1 ) + 4 l j ( l j − 1 ) − 4 ( p j + 2 ) ( n j − l j )     + 4 ( α j 2 − β j 2 ) ] + [ 4 n j ( n j − 1 ) s 2 2 ] − [ 4 l j ( l j − 1 ) t 2 2 ] − 8 n j ( α j ) s 2     − 8 l j ( β j ) t 2 tanh [ β j ( t 1 − t 2 ) ] + [ 4 α j ( 2 n j + 2 + p j ) ] s 2

  − [ p j ( n j + p j + l j + 1 ) s 2 2 u 2 ] + [ p j ( n j + p j + l j + 1 ) t 2 2 u 2 ]   − ( p j α j ) s 2 t 2 2 u 2 − ( p j β j ) t 2 s 2 2 u 2 tanh [ β j ( t 1 − t 2 ) ] + [ p j ( n j − l j ) s 1 2 u 2 ]   + [ p j ( n j − l j ) t 1 2 u 2 ] − ( p j α j ) s 2 s 1 2 u 2 − ( p j α j ) s 2 t 1 2 u 2

+ ( p j β j ) t 2 s 1 2 u 2 tanh [ β j ( t 1 − t 2 ) ] + ( p j β j ) t 2 t 1 2 u 2 tanh [ β j ( t 1 − t 2 ) ] − ( 2 p j β j ) s 1 s 2 t 1 u 2 tanh [ β j ( t 1 − t 2 ) ] + ( 2 p j l j ) s 1 s 2 t 1 t 2 u 2 + ( 2 p j α j ) t 1 s 1 t 2 u 2 − ( 2 p j n j ) t 1 s 1 t 2 s 2 u 2 } χ j , (91b)

Now, we deal with our system through the coordinates s 1 , s 2 , t 1 , t 2 , u , v rather than s a , s b , t a , t b , u , v . We notice from Equations (91) that T a and T b are obtained from T 1 and T 2 , respectively, by replacing s 1 , s 2 , t 1 , t 2 , u , v by s a , s b , t a , t b , u , v , respectively. Thus, T a χ j and T b χ j can be derived via (91a) and (91b), respectively, using the same rearrangement.

Finally, from (84), applying the potential energy operator to the j-th component of the wavefunction, Equation (89), we find

V χ j = 2 v [ ( 1 u + 1 ) − 4 s 1 s 1 2 − t 1 2 − 4 s 2 s 2 2 − t 2 2 ] χ j (91c)

The forms of the matrix elements required for RRVM or VT are quite complicated and will not be presented here.

In the present work a heteromolecule is defined as a molecule composed of an atom and its antiatom. Thus, heterohydrogen is the bound states of hydrogen and antihydrogen atoms ( [49] [50]) which corresponds to case I defined at Equation (4), with m₁ = m_a = m_e and m₂ = m_b = M and m_e is the electronic mass, taken equals 1/2 in Hartree a.u. The reduced masses, Equation (3), reduced to μ₁₂ = μ_ab = m_e and M is set to be infinite. The threshold energy is now given by E_thr = −1 Hartree = −27.2 eV. Thus, Equation (8b) state that ε(σ) = E(σ), and ω(σ) = W(σ). The optimization of the nonlinear parameters of the wavefunction provided us with α = 1.95, β = 0.87 and γ = 1.53 when the first five components of the wavefunction (see Table 1) are considered.

With these values, the convergence of the total and binding energies of the ground-state of the heterohydrogen was investigated. The binding energy obtained using 25 components of the wave function was found to be equal to −0.7476 eV. This result is in complete argument with the extension of the theorem of four-body systems. The stability of this molecule against dissociation to the lowest threshold state will be discussed in the next subsection.

The above investigations have been extended to other four-body systems for which case I, Equation (4), is valid and σ (=m/M) is changing between 0 and 1. In this case the stability of the systems is tested against dissociation into pairs of atoms and antiatoms possessing the same mass ratios (σ). The results of these investigations are displayed in Figure 4.

The Figure illustrates the monotonic behavior of the binding energy as a function of the mass ratio σ, which agrees completely with the prediction presented in Figure 2 and confirms the possible formation of the four-body systems.

The results of the preceding section encourage the search for other four-body systems with internal clusters supported by case II, Equations (5), i.e. searching for exotic molecules stable against dissociation into light and heavy pairs. The most interesting systems [51] are those in which a positronium (Ps) is bound to a protonium (Pn), true muonium (Mu) or pionium (A_2π) exotic atoms. As a matter of fact, Pn, Mu and A_2π have been synthesized very recently at CERN Laboratory (see [37] [38] [39] , respectively). Thus, in addition to the positronium, the lastly mentioned works confirm the possible construction of other three

exotic atoms composed of particles and antiparticles. It has been shown that Pn is a neutral boson with zero baryon number, mean lifetime τ ~ 1.0 × 10⁻⁶ s and ground state energy −12485.845 eV. On the other hand, the true muonium Mu is made up of two leptons, a muon, µ⁻, and an antimuon µ⁺. The binding energy of this exotic atom [38] is equal to −1407 eV (which is very close to the theoretical value −1406.024 eV). It is formed in triplet state (ortho-muonium) with lifetime 1.81 × 10⁻¹² s and in singlet state (para-muonium) with lifetime 0.602 × 10⁻¹² s eV. (True muonium should be distinguished from muonium atom which is a bound state composed of an electron and a positive muon). Nevertheless, it has been shown [39] that pionium, a bound state composed of two bosons; a pion, π⁺, and an antipion π⁻, possesses a binding energy equals −1860 eV (which is very close to the theoretical value ?1898.56 eV), and a lifetime 2.89 × 10⁻¹⁵ s.

The first interesting system is the one composed of e⁻, e⁺, p⁻ and p⁺. The lowest dissociation channel of this system is consisting of a positronium (Ps) and a protonium (Pn), with threshold energy E_thr = −12,492.28 eV. The rigorous mathematical proof of the existence of such system is given in Section 3.3 under the consideration of Equations (41a), (61) and (62). To show that the four particle system could form a bound state stable against dissociation to a pair of positronium and protonium exotic atoms, the Hamiltonian (1) and the wavefunction (89)-(90), are employed within the context of Rayleigh-Ritz’ variational method discussed in Section 2.2. A set of 25 components have been employed for developing the trial wavefunction. Diagonalization of the Hamiltonian and optimization of the nonlinear parameters provided us with (see Equations (8a, b), ω(σ) = 0.000075 and total energy = −12,493.217 eV, i.e. the four body system has a binding energy W(σ) = −0.9369 eV. This result indicates that the four body system e⁻, e⁺, p⁻ and p⁺ is stable against dissociation to a positronium and a protonium. Thus, on considering the results obtained in the preceding section we remark that H ¯ H is stable against dissociation of any kind. Replacing the proton mass in the previous investigation with 2m_p+ and 3m_p+ allowed us to test whether H ¯ D and H ¯ T , respectively, are stable against their dissociation to their lowest possible thresholds. The decisive answer to this question is demonstrated in Table 2 (rows number 6 and 7, respectively). Thus, the present investigations confirm, for the first time, that all Hydrogen-Antihydrogen Molecules [41] [50] [51] [52] are stable against dissociation of any kind. A conclusion which should end contradicted arguments raised in the literature mainly by authors who prefer the treatment of four-body systems via Born-Oppenheimer approximations (see for example [53] [54] [55] [56]).

Our second goal in this section is focused on the characteristics of the variational energies calculated for other four body systems. Again, our main interest here is to test their stabilities against dissociation to the lowest possible channels, i.e. the positronium-true-muonium and positronium-pionium channels, respectively. Quite elaborate investigations using the same basis set employed in the preceding studies have shown that the total variational ground state energies of

the two systems are −1447.161 eV and −1907.225 eV, respectively. Knowing that E_thr(Ps Mu) = −1412.824 eV and E_thr(Ps A_2π) = −1905.36 eV, we realize (see rows 3 and 4, in Table 2), that the binding energies of both systems are W(Ps Mu) = −2.7852 eV and W(Ps A_2π) = −1.8276 eV, i.e. the systems are bound against dissociation to light (Ps) and heavy (Mu or A_2π) quasi atoms. Consequently, the four-body systems e⁻ e⁺ µ⁻ µ⁺ and e⁻ e⁺ π⁻ π⁺, are stable against dissociation of any kind.

Remember that ω(σ) is the shift below the threshold which is defined through the relation E g = ( 1 + ω ( σ ) ) E t h , where E_th is the sum of the binding energies of the dissociated clusters 12 and ab and E_g stands for the total ground state energy of the corresponding four-body system. It is obvious that the inequality ω(σ) ≤ ω(1) is always fulfilled. Rows 6 and 7 contain information about the quasi molecules Ps-p⁻d⁺ and Ps-p⁻t⁺, respectively. The bottom raw in Table 2 contains the results of our investigations of the four-body system µ⁻ µ⁺ π⁻ π⁺. They show that the system is stable against dissociation into true muonium and pionium exotic atoms.

One of the curious puzzles of Astrophysics is the observation of strong γ-radiations inform of ball lightning at distance considered as the edges of our universe (Ashby and Whitehead [53] , see also [54] [55] [56] [57]), and the appearance of isotopic cosmic background spectrum of Gamma-rays above 1 MeV (David and Huges [58] , see also [59]). The formation of exotic atoms at CERN composted of particles and antiparticles as well as the possible formation of different four-body systems (Section 4) support the possible coexistence of matter and antimatter. Thus, suggesting the possible existence of an overlap area between the universe and antiuniverse. With regarding to the origin of our universe, it seems that matter and antimatter were attracted immediately after the big bang to different centers of gravity, a process which led to relatively rapid cooling and construction of not completely separated universe and antiuniverse. This argument could also explain a recent discovery that cosmic rays are mainly composed of particles and antiparticles [60] [61] [62] [63] .

The possible formation of Hydrogen-Antihydrogen Molecules allows us to propose the following scenario for implementation in cold fusion which is alternative to a suggestion made by Fleischmann and Pons [64] [65] . In our case (see also [66] [67]), we argue that if a thermalized beam of antihydrogens passes through a palladium sheet in which hydrogen (deuterium or tritium), atoms are localized, a bound-state (or quazimolecular structure) could be formed. Additionally, if the antihydrogen enters the Coulomb barrier of the localized atom, the system could collapse in two different channels, namely the annihilation and fusion channels yielding to huge amount of energy which could be considered as promising alternatives to nuclear energy sources based on nuclear fission and fusion processes.