Scattering of a particle with internal structure from a single slit: exact numerical solutions

Scattering of a quantum particle with internal structure is fundamentally different from that of a point particle and shows quantum effects such as the modification of transmission due to tunnelling and trapping of the particle. As in a preceding paper (Shore et al 2014 New J. Phys. 17 013046) we consider a model of a symmetric, rigid rotor travelling through an aperture in a thin but impenetrable screen which is perpendicular to both the direction of motion and the rotation axis. We determine the quantum mechanical properties of this two-dimensional geometrical model using a quasi one-dimensional scattering problem with unconventional boundaries. Our calculations rely on finding the Green's function, which has a direct connection to the scattering matrix. Evaluated on a discrete lattice the Hamiltonian is ‘dressed’ by a self-energy correction that takes into account the open boundary conditions in an exact way. We find that the passage through the aperture can be suppressed or enhanced as a result of the rotational motion. These effects manifest themselves through resonances in the transmission probability as a function of incident energy and symmetry of the incident wavefunction. We determine the density-of-states to reveal the mode structure of resonant states and to exhibit the lifetimes of temporary trapping within the aperture.


Introduction
In a previous article [1], referred hereafter as article I, we have considered the motion of a structured particle, idealized as a symmetric rigid rotor, passing through an aperture whose size is comparable to the particle. This is an important problem, because there are a number of experimental situations where the internal structure of a quantum mechanical object can play a role: for a list of references see [1]. On the other hand this is an interesting question on its own, as already in the classical picture the relevant coordinates are constrained in a nontrivial way, while in the quantum mechanical treatment this leads to a complicated entanglement of the translational and rotational degrees of freedom. Article I introduced an approximation that decoupled the translational and rotational motions thereby obtaining a set of one-dimensional Schrödinger equations for the hindered translation with effective potentials depending on the rotational mode. The present work removes the simplification made in that work, and provides a method that solves the two-dimensional scattering problem within numerical accuracy.
Our article is organized as follows. In section 2 we summarize the model analyzed in article I. We present the relevant Hamiltonian and the boundary conditions that will determine the wavefunction. Section 3 presents the quantum dynamics as a quasi one-dimensional scattering problem showing analogy to the motion of charge carriers in nanostructures [2]. We use here the technique of the S matrix and the Greenʼs function suited to a numerical solution without the approximations used in article I. We then devote section 4 to the discussion of the results obtained with this technique, paying particular attention to the density-of-states (DOS) and to the transmission probability and its dependence on the classical constraints. Finally section 5 summarizes our results.
In order to keep the article self-contained we place details in appendices. In appendix A we present a derivation of the so-called Fisher-Lee relation that connects elements of the Greenʼs function and the scattering matrix. Appendix B is devoted to the evaluation of the Greenʼs function. Finally, appendix C provides details about the discretization of the scattering problem.

The model: a symmetric rigid rotor constrained to a track
In this section we formulate the problem of the quantum mechanical scattering of a rotor from a single slit. Here we are rather brief and refer for a more detailed treatment to article I. As described in article I we consider the passage of a particle with rigid but orientable internal structure, expressible as rotation of a rigid symmetric rod, through a single aperture, in an impenetrable thin screen. The motion takes place in a plane perpendicular to the screen. For further simplification the centre-of-mass motion is constrained to be along the symmetry axis of the aperture. With this assumption we are dealing with a two-dimensional problem characterized by the centre-of-mass coordinate x and angular coordinate φ. (See frame a ( ) of figure 1 for coordinates and length scales.)

The two-dimensional Hamiltonian
The classical Hamiltonian for our model problem is the sum of translational and rotational kinetic energies x 2 2 where M is the total mass of the particle and  is the moment of inertia. We introduce a dimensionless massdistribution parameter κ to relate M and  2 thereby quantifying the distribution of rotor mass. In coordinate representation the quantum-mechanical Hamiltonian obtained from (1) becomes the differential operator and with the dimensionless, scaled parameters of coordinate s and energy ε (a) A symmetric rotor of radius a rotating in a plane, approaches an aperture of radius b in a thin screen. The centre-of-mass of the rotor is constrained to move along the symmetry axis of the aperture. This way the centre-of-mass is characterized by a single coordinate x measured along this axis. The other coordinate is the orientation of the rotor, described by the angle φ, with respect to this symmetry axis. The relevant parameter describing the geometrical situation is the aperture-to-length ratio ≡ c b a. The geometrical connection between the centre-of-mass coordinate and allowable rotation angles for a classical rotor passing through the aperture is demonstrated in frames b ( ) and c ( ), where we have used the dimensionless coordinate s defined by (4).
the energy eigenvalue equation is

Rotational constraints as boundary conditions
The eigenvalue equation (5) is to be completed by boundary conditions. We fix these by adapting the classical geometrical constraints. Classically the effect of the aperture is to hinder rotation as the particle approaches the screen. In frames b ( )and c ( ) of figure 1 we demonstrate the geometrical connection between the centre-of-mass coordinate and the allowable rotation angles.
To characterize the geometry of the rotor relative to the aperture we need, in addition to the parameters κ and ε, the aperture-to-length ratio As seen in figure 1, when the rotor is within the spatial range c c c 2 the rotation angle φ is constrained by the inequalities In the following sections we will refer to this range as the interaction region  ( ). In figure 2 we show the allowed domains of the angle φ determined by (8). To incorporate the hindered motion and the resulting entanglement of the s and φ variables into the quantum-mechanical description we impose the boundary condition that the wavefunction must vanish at the border of the geometrically allowed angular domain shown as thick red lines in figure 2: 0, for . (9) c c In addition the physical configuration of a symmetric rotor, as considered here, is unchanged if it is rotated by π. Due to this symmetry the energy eigenfunctions of the rotor must obey the symmetry relation In figure 2 we denote this π-periodicity by thick dashed blue lines.

Classical transmission probability
In order to bring out the similarities and differences between classical and quantum mechanical scattering, we consider an ensemble of non-rotating classical particles far away from the aperture, all with fixed initial energy ε and with an orientation angle φ distributed evenly on the interval π φ π − ⩽ ⩽ 2 2 . According to (8) from this stream of classical non-rotating particles only those pass through the aperture whose angle of orientation at = − s s c satisfies the inequality For all other angles the classical rotor will be reflected. Figure 2. The geometrically allowed (white) and forbidden (shaded) rotation angles φ (vertical axis) as stipulated by (8) in the special case ≡ = c b a 0.5. In the quantum mechanical treatment of section 3 we will prescribe a vanishing wavefunction on the boundary of the classically allowed region (continuous red line), and a π-periodicity in the φ-variable (thick dashed blue line). α c is defined in (11).
In this section we briefly review the theoretical background to be used in solving the scattering problem, such as the S matrix, the Greenʼs function, and their connection. For further characterization of the process the concept of the spectral function and the ensuing DOS are also defined.

Formulation of the scattering problem
We treat the passage of the rotor through the aperture as a quasi one-dimensional stationary scattering problem [3], in which an incident wave of fixed energy ε from the left encounters an obstructed passageway shown in figure 2, resulting in transmitted and reflected waves. To ensure this behaviour, we prescribe two additional asymptotic boundary conditions onΨ ε .
3.1.1. The free particle Outside of the interaction region  the rotor is unaffected by aperture-induced hindrance and the twodimensional Schrödinger equation (5) is separable into translational and rotational variables while the energy eigenvalue ε is the sum of translational and rotational contributions m m

( r o t )
The rotational wavefunctions ϕ m are either periodic or acquire a '−' sign when the system undergoes a π rotation (10). In this article we restrict our analysis to free particle wavefunctions with π periodic rotational eigenfunctions: is constrained to be a real number. This means that at a fixed total energy ε we have a restricted number of travelling modes given by 0 where⌊ ⌋denotes the integer part.

The asymptotic free domains
The incident wave coming from the left is a two-dimensional partial wave decomposition The quantum counterpart of the classical transmission probabilityT c given by (12) is the transmitted probability current over the incident one Thus to calculate the transmission probability (22) we need to know the S matrices S LL and S RL , respectively 6 .

The Green's function
The most efficient approach to solve the scattering problem involves the Greenʼs function [4], which according to [2] can be physically interpreted as a generalized S matrix describing the response at any point r due to an excitation at point ′ r . In coordinate representation the desired Greenʼs function is defined, as a solution of the equation The positive infinitesimal η with η ⟶ + 0 ensures that we obtain the retarded Greenʼs function, which contains only outgoing waves far to the left and right from , i.e. that ensures the physically acceptable boundary conditions given by (20) and (21). The Greenʼs function ε G depends on the scaled energy ε, which we have indicated by a subscript.
In the abstract Hilbert space notation of quantum mechanics the defining equation (24) of the Greenʼs function is an operator equation where H is the Hamiltonian of the problem. This form expresses the fact that ε G( ) is the inverse of the energy minus the Hamiltonian, but this operator inverse should be treated carefully, as ε − 1 H [ ]is singular at the eigenvalues of H.

Partition of the system
The difficulty of the present problem arises from the complicated boundary conditions inside , as apparent from figure 2 and expressed by (8) and (9). These make the allowed values of the translational coordinate s and the rotation angle φ depend on each other. It is this non-separability that obstructs an exact analytical treatment, although for specific types of low energy incident waves we obtained results in article I by means of analytic approximations.
Rather than introduce analytic simplification, we shall represent the Hamiltonian on a discrete lattice, by using lattice points where the wavefunction is declared to be nonzero. In this discrete representation (25) becomes a system of linear equations for ε G( ) which can be solved in principle by matrix inversion. The posed problem is periodic in φ but covers an infinite interval of s so the discrete representation space is still of infinite size. Note that by simply cutting the infinite discrete representation at some point and changing to 6 Many times in the literature, instead of the S matrix defined in (23), a modified unitary S matrix is used, describing the linear connection between the incoming and outgoing fluxes, i.e. probability current amplitudes. a finite-size Hamiltonian we would automatically introduce artificial closed boundaries and we would get the Greenʼs function of an isolated finite region.
To avoid the isolation, we partition the Hilbert space more carefully into 'left lead + box + right lead' as indicated in figure 3 and incorporate the effect of the leads in the domain of the box in an exact way [2]. We choose the rectangular box , which encloses the interaction region, in such way that along the left ( L ) and right ( R ) boundaries of the box the conditions of (20) and (21)  The idea of dividing the Hilbert space into 'inner' and 'outer' parts and deriving the effect of the 'outer parts' on the 'inner parts' has a long history in many branches of quantum physics [5][6][7][8].

Energy wavefunctions from the Green's function
We next outline how to calculate the wavefunction inside the box (arising from the incident wave) with help of the Greenʼs function, and how to connect the free left and right domains via the S matrix.

Greenʼs function in the box
By projecting equation (25) on the three domains (left lead, box and right lead) and solving for ε G( ), which when restricted to the box domain will be denoted by ε G ( ) BB , we find according to appendix B the relation The correction Σ ε ( )to the Hamiltonian H BB is known as the self-energy. In the work of Feshbach [5,6] and others this term is a non-Hermitian effective potential. On a discrete lattice this non-Hermitian correction can be calculated exactly with the semi-infinite leads on the left and right side (see (C.5)).

The wavefunction in the box
At this point we could proceed to the S matrix without explicit calculation of the wavefunction as is usually done. But for our specific problem the explicit form of the wavefunction inside  is also interesting. We calculate it from the incident wave (19) and the Greenʼs function inside the Box, using the fact that we have open boundary conditions forΨ ε at  L and  R which must be inherited by ε G( ) [9]. For details we refer to appendix A. With the help of these conditions we evaluate the resulting integral from Greenʼs theorem (A.3), giving the wavefunction inside the box as We stress that equation (27) is exact, free of any infinite summation or approximation, and it requires ε G( ) only in the finite domain  i.e. ε G ( ) BB . We note that equations of the Lippmann-Schwinger type [10] also express formallyΨ ε on the full two-dimensional plane [11].

Scattering matrix from the Greenʼs function
Taking only excitations which correspond to incident waves from the left, we relate elements of the S matrix directly to the Greenʼs function evaluated at the left ( L ) and right ( R ) boundaries of . To do so we introduce the notation qp q p q p Figure 3. We enclose the interaction region  in a rectangular box , outside of which the wavefunctionΨ φ ε s ( , ) is expressible as products of unhindered-rotor wave functions ϕ s ( ) m and companion plane waves ± e k s i m given by (20) and (21). Our goal is to obtain numerically exact solutions within the box using a discrete mesh of points.
where the index q and the index p on G can be either L or R. The desired relationship between the blocks of S and ε G( ) at the corresponding boundaries takes the form: .
(29) This relationship is sometimes referred to in the literature as the Fisher-Lee relation [2,12]. For completeness we outline a simple alternative derivation of equation (29) in appendix A.

Characterization of the resonances
Due to the presence of the self-energy Σ ε ( )the discrete energy spectrum of the isolated box becomes a continuous one, while the eigenstates broaden into resonances [11]. The characteristic features of such an energy spectrum, containing long-living resonant states, are embodied in the spectral function, in the densityof-states (DOS), and in the local density-of-states (LDOS). We now briefly review these concepts.

Lifetimes
To gain more insight into the physics of the Greenʼs function of the box we return for a moment to the energy eigenvalue equation with the effective Hamiltonian (26) . We cannot calculate the solutions of the modified eigenvalue problem as Σ ε ( )is itself energy dependent, but we see the double effect of the non-Hermitian correction compared to the eigenvalue equation of the isolated box. The Hermitian part of Σ ε ( )simply shifts the eigenenergies ε μ,0 of the isolated problem, while the anti-Hermitian part brings in an imaginary contribution, resulting in ,0 The imaginary part leads to an exponential decay of the absolute-value squared wavefunction during time evolution. In this sense γ μ 1 represents the lifetime of state Ψ μ B .

Spectral function
All the relevant information contained in the solutions of the eigenvalue equation (30) is also contained in the so-called spectral function of the box, defined as [13] ε ε ε ε arising from (30) with the energies given by (31), from (32) we obtain in the abstract Hilbert space notation the relation [2] ∑

Density-of-states (DOS)
The energy spectrum of the system is characterized by the DOS  ε ( ), defined as the trace of the spectral

Local density-of-states (LDOS)
The LDOS gives us more insight into the nature of the peaks in  ε ( ). This state-dependent function is defined as the spectral function (32) evaluated in coordinate representation for diagonal elements r in  In this limit the LDOS ρ ε r ( )provides the spatial distribution of the probability density of the resonant states.

Numerical results
In this section we present numerical solutions of the scattering problem posed in section 3 for several values of the aperture-to-length ratio and for different incident wave symmetries. Notable is the presence of scattering resonances, in which the rotor undergoes temporary trapping within the interaction region . Besides the calculation of the transmission probability (22) we also analyze this phenomenon with the aid of the concepts introduced in subsection 3.5.
We have carried out the calculation on a discrete lattice with 49 × 49 lattice points corresponding to a box  of size π π π π − × − [ 2, 2] [ 2,2]in the dimensionless coordinates s and φ. We chose this interval by refining the grid until we saw convergence in the results. The energy range investigated runs between 0 and E 50 0 , where κ ≡  E M a 2 ( ) 0 2 2 is the elementary unit of rotational excitation for our rotor. This range corresponds to 'cold' molecules, where we expect specific quantum effects.

Identification of the true resonances
According to (36) a plot of  ε ( )will reveal peaks corresponding to energies where long-living resonant states occur. For some of these peaks there is a concentration of probability (quasi-bound resonant states) within the region . Other peaks occur as a consequence of the dimensions of the box . We identify the latter by shifting the box  with respect to  along the s-axis. Such a shift will have little effect on the peaks corresponding to , while peaks due to the surrounding box will noticeably change.
Frame a ( )of figure 4 shows an example of the energy variation of the DOS  ε ( ). The plot reveals a series of peaks-resonances-as well as lesser variations. To clarify the nature of the resonances we have shifted the bounding box  along the s-axis to cover the interval π π Δ Δ − +n s s [ 2, 2] · [ , ]with = ⋯ n 0, 1, , 6. Frame b ( ) shows the resulting DOS variation. In this process out of the several peaks only the resonances denoted with solid lines did not change their position and height.
From this procedure we learn that out of the eight peaks shown in frame a ( )only six are characteristic of . We denote them as ε i , with or without a tilde. The energies ε i (ε i ) will turn out to correspond to even (odd) angular symmetry of the incident wavefunction, respectively. This question of symmetry is discussed in section 4.3.
We emphasize that the size and position of the box  along the s axis does not change the transmission properties of the system because the S matrix is invariant under the 'box-shifting' transformation. This gives an additional possibility to verify the precision and validity of the numerical procedure.

DOS and LDOS
Information about the wavefunctions corresponding to resonant energies comes from the LDOS ρ ε r ( ). Formulae (38) and (39) show that a long-living resonance has a LDOS proportional to the square of the probability distribution of that state, Ψ | | μ r ( ) B 2 . We expect that long-living states whose properties are insensitive to the size and location of the box  will be localized within the interaction region . Figure 5 verifies this expectation, indeed the first two peaks of  ε ( )are associated with dominant concentration of probability within . In contrast, panels corresponding to dotted energy values belong to probability concentration outside of  but within . This feature confirms that these energies are not characteristic of .
In order to exhibit energies at which long-living states appear, we present in figure 6 the DOS  ε ( )by a contour plot as a function both of scaled energy ε and of aperture-to-length parameter c. This picture extends the results shown in frame a ( )of figure 4 for a single value of c = 0.5 to a wide range of aperture-to-length parameter values. The white domains for lower c values (lower right part of the contour plot) indicate that for higher energies new long-living resonant states may appear. ( )only six are characteristic of . We denote them by ε i and ε i . At the resonances ε i the spatial symmetry of the wavefunction is even while at ε i it is odd. The second peak from the left, labelled ε 2 , is associated with a long-living resonant state of the system, as follows from its narrow width. From the height of the peak we deduce that the state has a lifetime more than ten times longer than the lifetime of any other state. Figure 5. Density-of-states  ε ( )as a function of scaled energy ε for aperture-to-length ratio c = 0.5 and mass-distribution parameter κ = 1. Above and below we show the density plots of the local density-of-states ρ ε r ( )for eight selected energies. For narrow resonances ρ ε r ( )is the probability density. The first two plots belonging to ε 1 and ε 2 display strong concentration within . The second peak is much sharper than all the other ones and its wavefunction exhibits a single horizontal node. The second and fourth upper panel, that correspond to energy values marked by dotted lines, exhibit concentration within  but outside of . This feature identifies these peaks as artefacts of the box . The other peaks show additional nodal structures in both s and φ, becoming more complicated with increasing energy.

Transmission for different rotational modes
In this section we examine how the transmission probability (22) depends on the choice of the incident wavefunction (19). In section 4.1 we used the DOS to identify the energies where resonances appear. Depending on the rotational symmetry of the incoming mode we can identify two essentially different types of transmission behaviour. Incident wavefunctions with odd or even symmetry in the φ variable will excite different resonance energies.
In order to reveal the genuine quantum effects in the scattering process, we introduce a slight generalization of the classical transmission probability T c defined by (12). We regard the absolute-value square of the φdependent part of the incident wavefunction (19) as a distribution of incoming angles for an ensemble of rotors. Accordingly we define the pseudo-classical transmission probability by the overlap between the incident distribution and the opening of the interaction region, as given by the integral k s k s 0 The example of (41), taking only the m = 0 term, corresponds to a non-rotating molecule, which was the main subject of article I. We note that in this special case the pseudo-classical transmission probability (40) is equal to the classical one (12). We present numerical results of the non-rotating case in figure 7. These justify the conclusions of article I and show the validity regime of the approximations used there. Exactly at the first resonant energy ε 1 identified from the DOS we see a peak in the transmission, as well. The position and character of this peak is well described by the analytic calculations in article I.
An other interesting effect is the transmission minimum, implying total reflection, at the third resonance energy ε 3 . Such minima are not predicted by the approximation of mode decoupling used in article I and that approximation must be abandoned as the energy increases. The second upper panel of figure 7 corresponding to ε 3 , gives a hint for the reason of no transmission. When coupled to the incident wave, the symmetry with respect to the s-axis gets deformed, the wavefunction vanishes at  R and nothing can get through to the right side. Figure 8 shows the comparison of the transmissions for incident waves (41) and (42), which exhibit the same overall characteristics. This statement also holds for other possible incident waves with even symmetry in φ, which suggests that the positions and the shapes of the peaks and dips in the transmission are characteristic of the incident wave symmetry with respect to φ.

Odd incident wave
Next we consider incident waves which are odd in the φ variable, which corresponds to the symmetry relation = − − c c m m in equation (19). The most elementary example in the energy regime ε < 50 reads Note that this incident wave is only possible for energies ε > 4, and this is the only possibility to create an odd incident-wave in the energy regime ε < ⩽ 4 16. Looking at the lower left panel in figure 5 we see that the long-living resonant state at energy ε 2 has just this odd symmetry. The effect of this resonant state appears in the transmission plot of figure 9 as a very narrow and high peak exactly at ε 2 . Otherwise there is almost no transmission for the odd incident wave (43) in the ε < < 4 20 energy range. From the probability-densities Ψ B 2 at ε 2 , shown in the first upper panel of figure 9, we recognize that with this odd superposition of incident modes it is possible to excite the long-living resonant state at ε 2 whose existence appeared in the DOS in figure 4. It is also possible to excite this state with one of the two rotating mode functions ϕ φ − ( ) 1 or ϕ φ ( ) 1 . However, the odd superposition given by (43) results in a significantly reduced overall transmission except for the resonance at ε 2 , which shows up therefore more sharply.
Enhancement of transmission appears also at ε 4 and ε 5 , the wavefunction for these energies are also shown in figure 9.

A true bound state of the system
In this section we examine the long-living resonance we have found for c = 0.5 at ε = 13.57 2 in the DOS (figure 5). In section 4.3.2 we demonstrated that this state causes a sharp peak in the transmission plot for incident waves with odd symmetry (figure 9). We shall now consider how the transmission peak varies with the aperture-to-length ratio c .
Indeed, the position of this resonance should depend on the value of the parameter c as apparent in figure 6. In figure 10 we show that the transmission maxima coincide perfectly the resonant energies deduced from the DOS. We also see that as the size of the opening gets smaller i.e. as c decreases the transmission is diminishing and finally below c = 0.27 no transmission is possible.
No transmission, however, does not mean that the corresponding resonant state can not be excited through the small opening by the incident wave. In figure 11 we show the absolute value square of the wavefunction inside  in case of the odd incident wave for c = 0.27. We see that the long-living state builds up exactly at the resonant-energy, although-as we said before-its existence is not visible in the transmission i.e. it does not couple to the right lead. Figure 12 shows the approximate lifetimes of this specific resonance as a function of c. We saw already in figure 4 for c = 0.5 that this state has a much longer lifetime than any other possible state.
The very narrow and finally disappearing transmission peaks and extremely long lifetimes suggest that around the parameter value c = 0.3 the corresponding state is a true bound state of the interaction region ().

Conclusions
In the present article we have investigated a model describing the stationary scattering problem of a rotating cold diatomic molecule passing through an aperture in a restricted two-dimensional geometry. We have calculated  the transmission and the reflection coefficients as functions of the energy for different rotating modes of the incoming molecule. The calculations were carried out by a numerical method based on the Greenʼs function of the problem in a discrete lattice representation. The complicated boundary conditions were taken into account by modifying the Hamiltonian with appropriate self-energy corrections.
We have found resonances in the DOS which could be classified according to the rotational symmetry of the incident wavefunction. Very sharp resonances, corresponding to long-living quasi-bound resonant states, appear. The energies and lifetimes of these resonant states depend on the aperture-to-length ratio c and the rotational symmetry properties of the incident wave.
As the aperture gets smaller than the size of the rotating molecule, the resonance peaks in the transmission get narrower and finally disappear while the lifetime of the corresponding excited state increases indefinitely, suggesting that we have found a true bound state of the system. Figure 11. Build-up of the long-living resonant state inside region  for the odd incident-wave of (43) for the aperture-to-length parameter value c = 0.27. At the resonant energy ε = 21.133 the corresponding wavefunction is located almost completely inside  and supports zero transmission probability.  Note that for both regions we have introduced overall phase factors ± e k D i n , which is equivalent to using different coordinate systems for the left and right leads having their origins at the respective ends of the leads. This change has no effect on the transmission and reflection coefficients containing only the absolute values of the r n and t n amplitudes.
According to the definition of the S matrix elements (23)   to describe the coupling between the interior region  and the semi-infinite external leads, and exploited the fact that H is Hermitian. During the discretization process we take these links to be a nearest neighbour coupling (see (C.4)) [11]. We also used the fact that the left lead and the right lead are not connected directly, but only through the box i.e.
We note that for a discrete lattice the dispersion relation is modified, the k m in (B.10) has to be replaced by Δ Δ = k k s s sin ( ) m m . In this way the finite size matrix to be inverted for G BB in (26) is fully defined.