code: def change_basis_real_to_complex(
      k: int,
      dtype: Optional[torch.dtype] = None,
      device: Optional[torch.device] = None,
  ) -> torch.Tensor:
      r"""Construct a transformation matrix to change the basis from real to complex spherical harmonics.
  
      This function constructs a transformation matrix Q that converts real spherical
      harmonics into complex spherical harmonics.
      It operates on the basis functions $Y_{\ell m}$ and $Y_{\ell}^{m}$, and accounts
      for the relationship between the real and complex forms of these harmonics
      as defined in the provided mathematical expressions.
  
      The resulting transformation matrix Q is used to change the basis of vectors or tensors of real spherical harmonics to
      their complex counterparts.
  
      Parameters
      ----------
      k : int
          The representation index, which determines the order of the representation.
      dtype : torch.dtype, optional
          The data type for the output tensor. If not provided, the
          function will infer it. Default is None.
      device : torch.device, optional
          The device where the output tensor will be placed. If not provided,
          the function will use the default device. Default is None.
  
      Returns
      -------
      torch.Tensor
          A transformation matrix Q that changes the basis from real to complex spherical harmonics.
  
      Notes
      -----
      Spherical harmonics Y_l^m are a family of functions that are defined on the surface of a
      unit sphere. They are used to represent various physical and mathematical phenomena that
      exhibit spherical symmetry. The indices l and m represent the degree and order of the
      spherical harmonics, respectively.
  
      The conversion from real to complex spherical harmonics is achieved by applying specific
      transformation coefficients to the real-valued harmonics. These coefficients are derived
      from the properties of spherical harmonics.
  
      References
      ----------
      .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics#Real_form
  
      Examples
      --------
      # The transformation matrix generated is used to change the basis of a vector of
      # real spherical harmonics with representation index 1 to complex spherical harmonics.
      >>> change_basis_real_to_complex(1)
      tensor([[-0.7071+0.0000j,  0.0000+0.0000j,  0.0000-0.7071j],
              [ 0.0000+0.0000j,  0.0000-1.0000j,  0.0000+0.0000j],
              [-0.7071+0.0000j,  0.0000+0.0000j,  0.0000+0.7071j]])
      """
      q = torch.zeros((2 * k + 1, 2 * k + 1), dtype=torch.complex128)
  
      # Construct the transformation matrix Q for m in range(-k, 0)
      for m in range(-k, 0):
          q[k + m, k + abs(m)] = 1 / 2**0.5
          q[k + m, k - abs(m)] = complex(-1j / 2**0.5)  # type: ignore
  
      # Set the diagonal elements for m = 0
      q[k, k] = 1
  
      # Construct the transformation matrix Q for m in range(1, k + 1)
      for m in range(1, k + 1):
          q[k + m, k + abs(m)] = (-1) ** m / 2**0.5
          q[k + m, k - abs(m)] = complex(1j * (-1) ** m / 2**0.5)  # type: ignore
  
      # Apply the factor of (-1j)**k to make the Clebsch-Gordan coefficients real
      q = (-1j) ** k * q
  
      # Handle dtype and device options
      if dtype is None:
          default_type = torch.empty(0).dtype
          if default_type == torch.float32:
              dtype = torch.complex64
          elif default_type == torch.float64:
              dtype = torch.complex128
      if device is None:
          device = torch.empty(0).device
  
      # Ensure the tensor is contiguous and on the specified device
      return q.to(
          dtype=dtype,
          device=device,
          copy=True,
          memory_format=torch.contiguous_format,
      )
  
file_path: deepchem/deepchem/utils/equivariance_utils.py
imports: import math
  from typing import Optional
  import torch
label_desc: Use image node_id 2 for calling a global function with example usage: change_basis_real_to_complex(k, dtype, device) and returns: q