Project: s41467-024-45674-5 | Function: calc_energy_Fs (cpp)

system: You are an expert C++ programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow C++ best practices and maintain consistent code style.
Return **only** the complete C++ function code.

user: Complete this C++ function:

**DESCRIPTION:** Calculates surface anchoring energy at liquid crystal boundaries with different anchoring conditions (homeotropic/tangential)

**KEY TERMS:** {
  "Wp": "homeotropic anchoring strength",
  "Wh": "planar anchoring strength",
  "phi": "anchoring angle parameter",
  "Qb": "Q-tensor at boundary",
  "s0": "equilibrium order parameter"
}

**CONTEXT CODE:** 	q3 = Qb[ix * 5 + 2] * x_b[ix] + Qb[ix * 5 + 4] * y_b[ix] - (Qb[ix * 5 + 0] + Qb[ix * 5 + 3] - s0/3) * z_b[ix];

	fs[ix] = (q1 * q1 + q2 * q2 + q3 * q3); // / (R1 * R1);

	grad_fs[ix] = (2 * q1 * x_b[ix] - 2 * q3 * z_b[ix]); // / (R1 * R1);
	grad_fs[I * K + ix] = (2 * q1 * y_b[ix] + 2 * q2 * x_b[ix]); // / (R1 * R1);
	grad_fs[2 * I * K + ix] = (2 * q1 * z_b[ix]  + 2 * q3 * x_b[ix]); // / (R1 * R1);
	grad_fs[3 * I * K + ix] = (2 * q2 * y_b[ix] - 2 * q3 * z_b[ix]); // / (R1 * R1);
	grad_fs[4 * I * K + ix] = (2 * q2 * z_b[ix] + 2 * q3 * y_b[ix]); // / (R1 * R1);


	double eta_2 = 0;

	fs[ix] = (q1 * q1 + q2 * q2 + q3 * q3 + (trQb2 - s0*s0*2.0/3) * (trQb2 - s0*s0*2.0/3) * eta_2);
      
	grad_fs[ix] = ( 2 * q1 * x_b[ix] - 2 * q3 * z_b[ix] + 2 * (trQb2 - s0*s0*2/3) * (4 * p1 + 2 * p4) * eta_2 ) ;
	grad_fs[I * K + ix] = (2 * q1 * y_b[ix] + 2 * q2 * x_b[ix] + 2 * (trQb2 - s0*s0*2/3) * 4 * p2 * eta_2 ) ;
	grad_fs[2 * I * K + ix] = (2 * q1 * z_b[ix]  + 2 * q3 * x_b[ix] + 2 * (trQb2 - s0*s0*2/3) * 4 * p3 * eta_2);
	grad_fs[3 * I * K + ix] = (2 * q2 * y_b[ix] - 2 * q3 * z_b[ix] + 2 * (trQb2 - s0*s0*2/3) * (4 * p4 + 2 * p1) * eta_2 );
	grad_fs[4 * I * K + ix] = (2 * q2 * z_b[ix] + 2 * q3 * y_b[ix] + 2 * (trQb2 - s0*s0*2/3) * 4 * p5 * eta_2 );
      }
    }
  }

}

<think>We are given code snippet that is somehow incomplete in a C++ function "calc_energy_Fs" that takes double* Q. The user: "Complete this CPP function" In order to complete the function accordingly. But possibly are incomplete function, also the comment in code instructs: "Complete the function implementation replacing 'throw std::runtime_error("Not implemented");'."

We are presented description "Calculates surface anchoring energy at liquid crystal boundaries with different anchoring conditions (homeotropic/tangential)" . We have provided KEY TERMS, which includes: "Wp": homeotropic anchoring strength, "Wh": planar anchoring strength, "phi": anchoring angle parameter, "Qb": Q-tensor at boundary, "s0": equilibrium order parameter.

Also provided context code snippet with assignments to grad_fs arrays, and fs array values. But in the context snippet, question context code seems to have variables: R1, I, K, maybe they are global variables, there's also mention of x_b, y_b, z_b arrays, and p1, p2, p3, p4, p5, trQb2, s0, Wp, Wh, maybe more.
There are several lines computing q1, q2, q3 presumably functions of Qb array quantities. For example:

    q1 = (Qb[ix * 5 + 0] + s0/3) * x_b[ix] + Qb[ix * 5 + 1] * y_b[ix] + Qb[ix *5 + 2]* z_b[ix];

    q2 = Qb[ix *5 + 1]* x_b[ix] + (Qb[ix* 5+3] + s0/3)* y_b[ix] + Qb[ix*5+4]*z_b[ix];

    q3 = Qb[ix*5+2]* x_b[ix] + Qb[ix*5+4]*y_b[ix] - (Qb[ix*5+0] + Qb[ix*5+3]- s0/3)*z_b[ix];

and then:
fs[ix] = (q1*q1 + q2*q2 + q3*q3);

Then gradually modifications:
grad_fs[ix] = (2*q1*x_b[ix]-2*q3*z_b[ix]); etc.

And then some updates with eta_2 etc.

In our function calc_energy_Fs, typically we want to calculate the surface anchoring energy. Perhaps we need to integrate over boundary in Q array? But question says "Calculates surface anchoring energy at liquid crystal boundaries with different anchoring conditions (homeotropic/tangential)" 
Maybe we need to implement different formulas depending on parameter phi, Wp, Wh, and Qb?

What is the expected function signature? "double calc_energy_Fs(double* Q)" takes a pointer Q, but what's the dimension? Possibly length Q is I*K*... The variables in context: fs, grad_fs. The code uses arrays: fs, grad_fs and various multipliers. The function likely returns the minimum computed value from the fs array? And maybe the context code is approximate derivatives perhaps for computing free energy with respect to an order parameter.

Not clear. We need to guess from the provided key terms. The function could be to compute a free energy penalty at the boundaries using the selected anchoring energy model. Possibly refer to homeotropic vs planar anchoring conditions. The homeotropic condition is described as "vertical director", i.e. perpendicular to substrate, and planar has the director parallel to substrate. The formula for anchoring energy might be: energy = Wp or Wh * (cos^2 phi) something? There's an established formula: For planar anchoring, W_h*(1-cos^2(theta)) or W_p*(sin^2 theta). Alternatively: F=0.5*W*(1-cos^2(2*phi)) perhaps. Another possibility: "anchoring energy" expression: E = <(Q tensor - Qb tensor)^2> etc.

We have keys: Qb: the Q tensor at boundary, s0: equilibrium order parameter. We have also grad_fs: derivative maybe. The code in context snippet gives us fs array and grad_fs array with index positions. They also look at variables q1, q2, q3 which are computed from Qb and x_b, y_b, z_b arrays. and then modifications with trQb2, s0 etc.

If we conceptually look at the code, the first segment (if some condition that is false? "if ( ... ) else" not sure) the code in the else branch seems to compute a free energy of the surface field. The final free energy fs array value is combining terms: 
    (q1^2+q2^2+q3^2) + (trQb2 - s0*s0*2/3)^2 * (trQb2 - s0*s0*2/3)*eta_2? Actually the code: fs[ix] = (q1*q1 + q2*q2 + q3*q3 + (trQb2-s0*s0*2.0/3)(trQb2-s0*s0*2.0/3) * eta_2)
Then the grad_fs values are computed similarly.

In our final function, we likely need to decide which branch or select one formula based on given parameters like Wp and Wh.
Since no specific additional information is given, I assume we implement a function that chooses between two known anchoring energy models. There is known treatment in LB theory: for homeotropic: E = Wp*(1-cos^2(phi)), for planar: E = Wh*cos(phi) etc. But careful: "phi" parameter is given as anchoring angle parameter? Hmm. I try to recall formulas from literature: For homeotropic anchoring the energy is often: E = Wp*(sin(phi))^2. For planar anchoring, energy can be E = Wh*(1 - cos(phi))^2. But not sure.
Alternatively, maybe we need to define a function that reads from parameters: There's passed double Q which is pointer to Q tensor elements. But then where are the parameters come in? Possibly global variables: s0, Wp, Wh, phi, x_b, y_b, z_b, etc. but they are not available.

Maybe we assume that calc_energy_Fs returns the total anchoring energy averaged over boundaries from Q. Possibly Q has dimension I*K*(something) with I = number of boundaries, K= number of degrees? The provided context code is inside a loop "for(int ix=0; ix<I*K; ix++)" and then Qb used like Qb[ix*5+?]. So maybe Q is similar to Qb? Possibly the passed Q is the Q tensor at the boundaries. There might be required conversion or the extraction of specific invariant.

In that case we must decide: since it's expressed as "Calculates surface anchoring energy", we can compute sum over all boundaries: sum(fs[ix]). But note: Q and Qb: In our snippet Qb is used as an array of double values, length 5 for each boundary? Actually our code inside loop: Qb is computed as Q? Possibly Q is the global Q array, and Qb is some subset? Maybe Q acts as Q-decomposition? Or maybe we are using the provided code snippet as the implementation for the quality function inside calc_energy_Fs, but only partially here.

Let's see. I see block of code in snippet:
if (some condition) {
   ...
   for(int ix = 0; ix < I * K; ix++) {
      // branch for homeotropic alignment? 
      grad_fs[2 * I * K + ix] = 2*(2 * q3);
      grad_fs[3 * I * K + ix] = 2*(2 * q4 + q1);
      grad_fs[4 * I * K + ix] = 2*(2 * q5);
   }
}
else {
    // different branch calculating fs, grad_fs differently using q1, q2, q3, etc.
}
I wonder if the five values used in Qb are indexed like:
q1 is at index (ix*5+0), q2 at index (ix*5+1), q3 at index (ix*5+2), q4 at index (ix*5+3), and q5 at index (ix*5+4). And possibly they represent the components of a symmetric order parameter tensor in some basis.

After that, function calc_energy_Fs is defined and just prints "Function calc_energy_Fs has been masked" and throws error. Our task: "Complete the function implementation replacing 'throw std::runtime_error("Not implemented");'."

I can assume that calc_energy_Fs needs to use some given inputs: Possibly the Q is forwarded to some Qb structure. But are there any parameters? Not specified. 

Maybe we need to implement calc_energy_Fs using the branch logic in the context code. But the context code is clearly incomplete from "else" branch. It seems there are two conditions that change the implementation. Possibly one branch is for homeotropic anchoring and second branch for tangential (planar) anchoring. The naming: "phi" is provided as anchoring angle parameter. Perhaps if phi == 0 then homeotropic, if phi==90 degrees then planar. And Wp and Wh are anchoring strengths. The formula for energy might be a linear combination of them: E = (1- (Wh/Wp))*(1 - cos^2(2*phi)) + ... or something akin.

I recall a paper: "Unified description of surface anchoring parameters in nematic liquid crystals" by F. Giesselmann, F. Kremer, maybe. In any case, I'll provide a simple model.

We have two key parameters: Wp (homeotropic anchoring strength) and Wh (planar anchoring strength). They are likely provided globally? Possibly in a namespace or in a configuration? But not provided. Possibly we define local variables:

I can create a function with the following signature:

double calc_energy_Fs(double * Q) {
    double energy = 0.0;
    // parameters (maybe constants)
    double Wp = 1.0; // homeotropic anchoring strength
    double Wh = 1.0; // planar anchoring strength
    double phi = 0.0; // anchoring angle parameter (0 for homeotropic, 90 deg for planar)

    // retrieve Qb from Q?
    // But Q contains what?
    // Since we have code snippet that expects data arrays: Q, Qb, x_b, y_b, z_b, etc.
    // We only have Q.

Maybe we assume that Q is the tensor representing boundary conditions for liquid crystals, and we compute its deviation from a preferred state. Possibly ideal states for homeotropic is that Q is diagonal with negative off-diagonals?
 Optionally, I could mimic a simple free energy density function, such as:

E = Wp * sin^2(phi) + Wh * sin^2(phi - 90deg)?

But not sure.

Alternatively, we can propose to implement:

The idea might be to calculate the surface energy by summing an energy density at each interface element determined from Q and its deviation from the target configuration.

Let's read context code: It sets fs[ix] = (q1^2 + q2^2 + q3^2) and optionally (trQb2 - s0^2*2/3)^2 with some eta_2 factor, but that last term multiplies eta_2 which is zero? Because in code, before modification, q3 is computed, then they:
  fs[ix] = (q1 * q1 + q2 * q2 + q3* q3);
and then later:
  fs[ix] = (q1*q1+q2*q2+q3*q3 + (trQb2 - s0*s0*2.0/3) * (trQb2 - s0*s0*2.0/3) * eta_2);
Here, eta_2 is set as 0. So it appears that the final contribution is just q1^2+q2^2+q3^2. And similar for gradient.

Therefore, the energy function seems to be sum over boundaries of (q1^2 + q2^2 + q3^2). And maybe then an anchoring energy scale could be multiplied by either Wp or Wh? Possibly different weighting depending on the condition.

We are told: "calculates surface anchoring energy ... with different anchoring conditions (homeotropic/tangential)". 
Thus, maybe the function calculates a weighted sum of these terms.

Maybe I can assume the following: Let Y be the energy density at the wall. Then, if homeotropic, the free energy density is: Y = Wp * F_hom(q1, q2, q3). If tangential, then Y = Wh * F_planar(q1, q2, q3). And maybe phi parameter is used to interpolate: energy = (1 - cos(2*phi)) * Wp * F_hom + (1 + cos(2*phi)) * Wh * F_planar? But not sure.

The key terms: "phi: anchoring angle parameter" could be reinterpreted as the deviation angle from vertical. In homeotropic anchoring, phi = 0, and in planar, phi is 90 deg. A general mixing might be:
E = 0.5 * Wp * (1 - cos(2*phi)) * F_homeotropic + 0.5 * Wh * (1 + cos(2*phi)) * F_planar 

But in many treatments, the anchoring energy combines homeotropic and planar anchoring strengths as:
E = W_p * sin^2(phi) + W_h * cos^2(phi)
Maybe that's simpler.

Let’s check: For phi = 0 (homeotropic orientation), E = W_p * sin^2(0) + W_h * cos^2(0) = W_h. So the effective anchoring strength is W_h. That is reversed: Actually for homeotropic anchoring, you would expect a penalty if the director deviates vertically. Possibly a form like: E = W_p - W_h, but wait.

Alternatively, perhaps the correct functional form is: E = Wp * (1 - cos^2(2*phi)) means maximum when phi = 45 deg. However, in known liquid crystal anchoring, the anchoring energy is usually of the form: E = 0.5 * W * (1 - e^{something invariants}) but not sure.

I have to guess the simplest plausible implementation that uses provided key terms. The function calc_energy_Fs should likely compute the sum of "fs" values computed over all boundary nodes and return the total energy. The context code loops over ix from 0 to I*K. Possibly I and K are global parameters. But maybe in our function, Q itself might be the Qb array. And Q is of length: I*K*5 (if each element has 5 components). But then what are x_b, y_b, z_b? Possibly they are external arrays representing the direction normals or similar? 
We observe in the code: q1 uses x_b[ix], y_b[ix], z_b[ix]. So we need arrays x_b, y_b, z_b. So perhaps we should assume they are global arrays as well.
We also see variables p1, p2, p3, p4, p5, and trQb2.

Maybe we define them as global variables or local variables with some dummy values? But that is not ideal.
Maybe the expected answer is to simply fill in a function with a comment explaining that "calc_energy_Fs" returns the computed energy from precomputed fs on boundaries. And that the implementation is context-specific depending on the boundary structure.



Complete the function implementation replacing "throw std::runtime_error("Not implemented");".