def mgga_sum_(vmat, ao, wv, mask):
aow = numint._scale_ao(ao[:4], wv[:4])
tmp = numint._dot_ao_ao(mol, ao[0], aow, mask, shls_slice, ao_loc)
aow = numint._scale_ao(ao[1], wv[5], aow)
tmp += numint._dot_ao_ao(mol, ao[1], aow, mask, shls_slice, ao_loc)
aow = numint._scale_ao(ao[2], wv[5], aow)
tmp += numint._dot_ao_ao(mol, ao[2], aow, mask, shls_slice, ao_loc)
aow = numint._scale_ao(ao[3], wv[5], aow)
tmp += numint._dot_ao_ao(mol, ao[3], aow, mask, shls_slice, ao_loc)
vmat[0] += tmp + tmp.T
rks_grad._gga_grad_sum_(vmat[1:], mol, ao, wv[:4], mask, ao_loc)
rks_grad._tau_grad_dot_(vmat[1:], mol, ao, wv[5] * 2, mask, ao_loc,
True)
def _get_vxc_deriv1(hessobj, mo_coeff, mo_occ, max_memory):
mol = hessobj.mol
mf = hessobj.base
if hessobj.grids is not None:
grids = hessobj.grids
else:
grids = mf.grids
if grids.coords is None:
grids.build(with_non0tab=True)
nao, nmo = mo_coeff.shape
ni = mf._numint
xctype = ni._xc_type(mf.xc)
aoslices = mol.aoslice_by_atom()
shls_slice = (0, mol.nbas)
ao_loc = mol.ao_loc_nr()
dm0 = mf.make_rdm1(mo_coeff, mo_occ)
v_ip = numpy.zeros((3, nao, nao))
vmat = numpy.zeros((mol.natm, 3, nao, nao))
max_memory = max(2000, max_memory - vmat.size * 8 / 1e6)
if xctype == 'LDA':
ao_deriv = 1
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[0], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc(mf.xc, rho, 0, deriv=2)[1:3]
vrho = vxc[0]
frr = fxc[0]
aow = numint._scale_ao(ao[0], weight * vrho)
rks_grad._d1_dot_(v_ip, mol, ao[1:4], aow, mask, ao_loc, True)
ao_dm0 = numint._dot_ao_dm(mol, ao[0], dm0, mask, shls_slice,
ao_loc)
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
# First order density = rho1 * 2. *2 is not applied because + c.c. in the end
rho1 = numpy.einsum('xpi,pi->xp', ao[1:, :, p0:p1],
ao_dm0[:, p0:p1])
wv = weight * frr * rho1
aow = [numint._scale_ao(ao[0], wv[i]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[0], mask, ao_loc,
True)
ao_dm0 = aow = None
elif xctype == 'GGA':
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:4], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc(mf.xc, rho, 0, deriv=2)[1:3]
wv = numint._rks_gga_wv0(rho, vxc, weight)
rks_grad._gga_grad_sum_(v_ip, mol, ao, wv, mask, ao_loc)
ao_dm0 = [
numint._dot_ao_dm(mol, ao[i], dm0, mask, shls_slice, ao_loc)
for i in range(4)
]
for ia in range(mol.natm):
wv = dR_rho1 = _make_dR_rho1(ao, ao_dm0, ia, aoslices)
wv[0] = numint._rks_gga_wv1(rho, dR_rho1[0], vxc, fxc, weight)
wv[1] = numint._rks_gga_wv1(rho, dR_rho1[1], vxc, fxc, weight)
wv[2] = numint._rks_gga_wv1(rho, dR_rho1[2], vxc, fxc, weight)
aow = [numint._scale_ao(ao[:4], wv[i, :4]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[0], mask, ao_loc,
True)
ao_dm0 = aow = None
elif xctype == 'MGGA':
if grids.level < 5:
logger.warn(mol, 'MGGA Hessian is sensitive to dft grids.')
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho = ni.eval_rho2(mol, ao[:10], mo_coeff, mo_occ, mask, xctype)
vxc, fxc = ni.eval_xc(mf.xc, rho, 0, deriv=2)[1:3]
wv = numint._rks_mgga_wv0(rho, vxc, weight)
rks_grad._gga_grad_sum_(v_ip, mol, ao, wv, mask, ao_loc)
# *2 because wv[5] is scaled by 0.5 in _rks_mgga_wv0
rks_grad._tau_grad_dot_(v_ip, mol, ao, wv[5] * 2, mask, ao_loc,
True)
ao_dm0 = [
numint._dot_ao_dm(mol, ao[i], dm0, mask, shls_slice, ao_loc)
for i in range(4)
]
for ia in range(mol.natm):
wv = dR_rho1 = _make_dR_rho1(ao, ao_dm0, ia, aoslices, xctype)
wv[0] = numint._rks_mgga_wv1(rho, dR_rho1[0], vxc, fxc, weight)
wv[1] = numint._rks_mgga_wv1(rho, dR_rho1[1], vxc, fxc, weight)
wv[2] = numint._rks_mgga_wv1(rho, dR_rho1[2], vxc, fxc, weight)
aow = [numint._scale_ao(ao[:4], wv[i, :4]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[0], mask, ao_loc,
True)
for j in range(1, 4):
aow = [numint._scale_ao(ao[j], wv[i, 5]) for i in range(3)]
rks_grad._d1_dot_(vmat[ia], mol, aow, ao[j], mask, ao_loc,
True)
ao_dm0 = aow = None
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
vmat[ia, :, p0:p1] += v_ip[:, p0:p1]
vmat[ia] = -vmat[ia] - vmat[ia].transpose(0, 2, 1)
return vmat
def get_vxc(ni,
mol,
grids,
xc_code,
dms,
relativity=0,
hermi=1,
max_memory=2000,
verbose=None):
xctype = ni._xc_type(xc_code)
make_rho, nset, nao = ni._gen_rho_evaluator(mol, dms, hermi)
ao_loc = mol.ao_loc_nr()
vmat = numpy.zeros((2, 3, nao, nao))
if xctype == 'LDA':
ao_deriv = 1
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho_a = make_rho(0, ao[0], mask, xctype)
rho_b = make_rho(1, ao[0], mask, xctype)
vxc = ni.eval_xc(xc_code, (rho_a, rho_b),
1,
relativity,
1,
verbose=verbose)[1]
vrho = vxc[0]
#:aow = numpy.einsum('pi,p->pi', ao[0], weight*vrho[:,0])
aow = numint._scale_ao(ao[0], weight * vrho[:, 0])
rks_grad._d1_dot_(vmat[0], mol, ao[1:4], aow, mask, ao_loc, True)
#:aow = numpy.einsum('pi,p->pi', ao[0], weight*vrho[:,1])
aow = numint._scale_ao(ao[0], weight * vrho[:, 1])
rks_grad._d1_dot_(vmat[1], mol, ao[1:4], aow, mask, ao_loc, True)
elif xctype == 'GGA':
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho_a = make_rho(0, ao[:4], mask, xctype)
rho_b = make_rho(1, ao[:4], mask, xctype)
vxc = ni.eval_xc(xc_code, (rho_a, rho_b),
1,
relativity,
1,
verbose=verbose)[1]
wva, wvb = numint._uks_gga_wv0((rho_a, rho_b), vxc, weight)
rks_grad._gga_grad_sum_(vmat[0], mol, ao, wva, mask, ao_loc)
rks_grad._gga_grad_sum_(vmat[1], mol, ao, wvb, mask, ao_loc)
elif xctype == 'NLC':
raise NotImplementedError('NLC')
elif xctype == 'MGGA':
ao_deriv = 2
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rho_a = make_rho(0, ao[:10], mask, xctype)
rho_b = make_rho(1, ao[:10], mask, xctype)
vxc = ni.eval_xc(xc_code, (rho_a, rho_b),
1,
relativity,
1,
verbose=verbose)[1]
wva, wvb = numint._uks_mgga_wv0((rho_a, rho_b), vxc, weight)
rks_grad._gga_grad_sum_(vmat[0], mol, ao, wva, mask, ao_loc)
rks_grad._gga_grad_sum_(vmat[1], mol, ao, wvb, mask, ao_loc)
# *2 because wv[5] is scaled by 0.5 in _uks_mgga_wv0
rks_grad._tau_grad_dot_(vmat[0], mol, ao, wva[5] * 2, mask, ao_loc,
True)
rks_grad._tau_grad_dot_(vmat[1], mol, ao, wvb[5] * 2, mask, ao_loc,
True)
exc = numpy.zeros((mol.natm, 3))
# - sign because nabla_X = -nabla_x
return exc, -vmat
def _get_vxc_deriv1(hessobj, mo_coeff, mo_occ, max_memory):
mol = hessobj.mol
mf = hessobj.base
if hessobj.grids is not None:
grids = hessobj.grids
else:
grids = mf.grids
if grids.coords is None:
grids.build(with_non0tab=True)
nao, nmo = mo_coeff[0].shape
ni = mf._numint
xctype = ni._xc_type(mf.xc)
aoslices = mol.aoslice_by_atom()
shls_slice = (0, mol.nbas)
ao_loc = mol.ao_loc_nr()
dm0a, dm0b = mf.make_rdm1(mo_coeff, mo_occ)
vmata = numpy.zeros((mol.natm, 3, nao, nao))
vmatb = numpy.zeros((mol.natm, 3, nao, nao))
max_memory = max(2000, max_memory - (vmata.size + vmatb.size) * 8 / 1e6)
if xctype == 'LDA':
ao_deriv = 1
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rhoa = ni.eval_rho2(mol, ao[0], mo_coeff[0], mo_occ[0], mask,
xctype)
rhob = ni.eval_rho2(mol, ao[0], mo_coeff[1], mo_occ[1], mask,
xctype)
vxc, fxc = ni.eval_xc(mf.xc, (rhoa, rhob), 1, deriv=2)[1:3]
vrho = vxc[0]
u_u, u_d, d_d = fxc[0].T
ao_dm0a = numint._dot_ao_dm(mol, ao[0], dm0a, mask, shls_slice,
ao_loc)
ao_dm0b = numint._dot_ao_dm(mol, ao[0], dm0b, mask, shls_slice,
ao_loc)
aow1a = numpy.einsum('xpi,p->xpi', ao[1:], weight * vrho[:, 0])
aow1b = numpy.einsum('xpi,p->xpi', ao[1:], weight * vrho[:, 1])
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
# First order density = rho1 * 2. *2 is not applied because + c.c. in the end
rho1a = numpy.einsum('xpi,pi->xp', ao[1:, :, p0:p1],
ao_dm0a[:, p0:p1])
rho1b = numpy.einsum('xpi,pi->xp', ao[1:, :, p0:p1],
ao_dm0b[:, p0:p1])
wv = u_u * rho1a + u_d * rho1b
wv *= weight
aow = numpy.einsum('pi,xp->xpi', ao[0], wv)
aow[:, :, p0:p1] += aow1a[:, :, p0:p1]
rks_grad._d1_dot_(vmata[ia], mol, aow, ao[0], mask, ao_loc,
True)
wv = u_d * rho1a + d_d * rho1b
wv *= weight
aow = numpy.einsum('pi,xp->xpi', ao[0], wv)
aow[:, :, p0:p1] += aow1b[:, :, p0:p1]
rks_grad._d1_dot_(vmatb[ia], mol, aow, ao[0], mask, ao_loc,
True)
ao_dm0a = ao_dm0b = aow = aow1a = aow1b = None
for ia in range(mol.natm):
vmata[ia] = -vmata[ia] - vmata[ia].transpose(0, 2, 1)
vmatb[ia] = -vmatb[ia] - vmatb[ia].transpose(0, 2, 1)
elif xctype == 'GGA':
ao_deriv = 2
vipa = numpy.zeros((3, nao, nao))
vipb = numpy.zeros((3, nao, nao))
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rhoa = ni.eval_rho2(mol, ao[:4], mo_coeff[0], mo_occ[0], mask,
xctype)
rhob = ni.eval_rho2(mol, ao[:4], mo_coeff[1], mo_occ[1], mask,
xctype)
vxc, fxc = ni.eval_xc(mf.xc, (rhoa, rhob), 1, deriv=2)[1:3]
wva, wvb = numint._uks_gga_wv0((rhoa, rhob), vxc, weight)
rks_grad._gga_grad_sum_(vipa, mol, ao, wva, mask, ao_loc)
rks_grad._gga_grad_sum_(vipb, mol, ao, wvb, mask, ao_loc)
ao_dm0a = [
numint._dot_ao_dm(mol, ao[i], dm0a, mask, shls_slice, ao_loc)
for i in range(4)
]
ao_dm0b = [
numint._dot_ao_dm(mol, ao[i], dm0b, mask, shls_slice, ao_loc)
for i in range(4)
]
for ia in range(mol.natm):
wva = dR_rho1a = rks_hess._make_dR_rho1(
ao, ao_dm0a, ia, aoslices)
wvb = dR_rho1b = rks_hess._make_dR_rho1(
ao, ao_dm0b, ia, aoslices)
wva[0], wvb[0] = numint._uks_gga_wv1(
(rhoa, rhob), (dR_rho1a[0], dR_rho1b[0]), vxc, fxc, weight)
wva[1], wvb[1] = numint._uks_gga_wv1(
(rhoa, rhob), (dR_rho1a[1], dR_rho1b[1]), vxc, fxc, weight)
wva[2], wvb[2] = numint._uks_gga_wv1(
(rhoa, rhob), (dR_rho1a[2], dR_rho1b[2]), vxc, fxc, weight)
aow = [numint._scale_ao(ao[:4], wva[i, :4]) for i in range(3)]
rks_grad._d1_dot_(vmata[ia], mol, aow, ao[0], mask, ao_loc,
True)
aow = [numint._scale_ao(ao[:4], wvb[i, :4]) for i in range(3)]
rks_grad._d1_dot_(vmatb[ia], mol, aow, ao[0], mask, ao_loc,
True)
ao_dm0a = ao_dm0b = aow = None
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
vmata[ia, :, p0:p1] += vipa[:, p0:p1]
vmatb[ia, :, p0:p1] += vipb[:, p0:p1]
vmata[ia] = -vmata[ia] - vmata[ia].transpose(0, 2, 1)
vmatb[ia] = -vmatb[ia] - vmatb[ia].transpose(0, 2, 1)
elif xctype == 'MGGA':
if grids.level < 5:
logger.warn(mol, 'MGGA Hessian is sensitive to dft grids.')
ao_deriv = 2
vipa = numpy.zeros((3, nao, nao))
vipb = numpy.zeros((3, nao, nao))
for ao, mask, weight, coords \
in ni.block_loop(mol, grids, nao, ao_deriv, max_memory):
rhoa = ni.eval_rho2(mol, ao[:10], mo_coeff[0], mo_occ[0], mask,
xctype)
rhob = ni.eval_rho2(mol, ao[:10], mo_coeff[1], mo_occ[1], mask,
xctype)
vxc, fxc = ni.eval_xc(mf.xc, (rhoa, rhob), 1, deriv=2)[1:3]
wva, wvb = numint._uks_mgga_wv0((rhoa, rhob), vxc, weight)
rks_grad._gga_grad_sum_(vipa, mol, ao, wva, mask, ao_loc)
rks_grad._gga_grad_sum_(vipb, mol, ao, wvb, mask, ao_loc)
rks_grad._tau_grad_dot_(vipa, mol, ao, wva[5] * 2, mask, ao_loc,
True)
rks_grad._tau_grad_dot_(vipb, mol, ao, wvb[5] * 2, mask, ao_loc,
True)
ao_dm0a = [
numint._dot_ao_dm(mol, ao[i], dm0a, mask, shls_slice, ao_loc)
for i in range(4)
]
ao_dm0b = [
numint._dot_ao_dm(mol, ao[i], dm0b, mask, shls_slice, ao_loc)
for i in range(4)
]
for ia in range(mol.natm):
wva = dR_rho1a = rks_hess._make_dR_rho1(
ao, ao_dm0a, ia, aoslices, xctype)
wvb = dR_rho1b = rks_hess._make_dR_rho1(
ao, ao_dm0b, ia, aoslices, xctype)
wva[0], wvb[0] = numint._uks_mgga_wv1(
(rhoa, rhob), (dR_rho1a[0], dR_rho1b[0]), vxc, fxc, weight)
wva[1], wvb[1] = numint._uks_mgga_wv1(
(rhoa, rhob), (dR_rho1a[1], dR_rho1b[1]), vxc, fxc, weight)
wva[2], wvb[2] = numint._uks_mgga_wv1(
(rhoa, rhob), (dR_rho1a[2], dR_rho1b[2]), vxc, fxc, weight)
aow = [numint._scale_ao(ao[:4], wva[i, :4]) for i in range(3)]
rks_grad._d1_dot_(vmata[ia], mol, aow, ao[0], mask, ao_loc,
True)
aow = [numint._scale_ao(ao[:4], wvb[i, :4]) for i in range(3)]
rks_grad._d1_dot_(vmatb[ia], mol, aow, ao[0], mask, ao_loc,
True)
for j in range(1, 4):
aow = [
numint._scale_ao(ao[j], wva[i, 5]) for i in range(3)
]
rks_grad._d1_dot_(vmata[ia], mol, aow, ao[j], mask, ao_loc,
True)
aow = [
numint._scale_ao(ao[j], wvb[i, 5]) for i in range(3)
]
rks_grad._d1_dot_(vmatb[ia], mol, aow, ao[j], mask, ao_loc,
True)
for ia in range(mol.natm):
p0, p1 = aoslices[ia][2:]
vmata[ia, :, p0:p1] += vipa[:, p0:p1]
vmatb[ia, :, p0:p1] += vipb[:, p0:p1]
vmata[ia] = -vmata[ia] - vmata[ia].transpose(0, 2, 1)
vmatb[ia] = -vmatb[ia] - vmatb[ia].transpose(0, 2, 1)
return vmata, vmatb
def get_vxc_full_response(ni,
mol,
grids,
xc_code,
dms,
relativity=0,
hermi=1,
max_memory=2000,
verbose=None):
'''Full response including the response of the grids'''
xctype = ni._xc_type(xc_code)
make_rho, nset, nao = ni._gen_rho_evaluator(mol, dms, hermi)
ao_loc = mol.ao_loc_nr()
aoslices = mol.aoslice_by_atom()
excsum = 0
vmat = numpy.zeros((2, 3, nao, nao))
if xctype == 'LDA':
ao_deriv = 1
for atm_id, (coords, weight, weight1) \
in enumerate(rks_grad.grids_response_cc(grids)):
sh0, sh1 = aoslices[atm_id][:2]
mask = gen_grid.make_mask(mol, coords)
ao = ni.eval_ao(mol, coords, deriv=ao_deriv, non0tab=mask)
rho_a = make_rho(0, ao[0], mask, xctype)
rho_b = make_rho(1, ao[0], mask, xctype)
exc, vxc = ni.eval_xc(xc_code, (rho_a, rho_b),
1,
relativity,
1,
verbose=verbose)[:2]
vrho = vxc[0]
vtmp = numpy.zeros((3, nao, nao))
#:aow = numpy.einsum('pi,p->pi', ao[0], weight*vrho[:,0])
aow = numint._scale_ao(ao[0], weight * vrho[:, 0])
rks_grad._d1_dot_(vtmp, mol, ao[1:4], aow, mask, ao_loc, True)
vmat[0] += vtmp
excsum += numpy.einsum('r,r,nxr->nx', exc, rho_a + rho_b, weight1)
excsum[atm_id] += numpy.einsum('xij,ji->x', vtmp, dms[0]) * 2
vtmp = numpy.zeros((3, nao, nao))
#:aow = numpy.einsum('pi,p->pi', ao[0], weight*vrho[:,1])
aow = numint._scale_ao(ao[0], weight * vrho[:, 1])
rks_grad._d1_dot_(vtmp, mol, ao[1:4], aow, mask, ao_loc, True)
vmat[1] += vtmp
excsum[atm_id] += numpy.einsum('xij,ji->x', vtmp, dms[1]) * 2
elif xctype == 'GGA':
ao_deriv = 2
for atm_id, (coords, weight, weight1) \
in enumerate(rks_grad.grids_response_cc(grids)):
sh0, sh1 = aoslices[atm_id][:2]
mask = gen_grid.make_mask(mol, coords)
ao = ni.eval_ao(mol, coords, deriv=ao_deriv, non0tab=mask)
rho_a = make_rho(0, ao[:4], mask, xctype)
rho_b = make_rho(1, ao[:4], mask, xctype)
exc, vxc = ni.eval_xc(xc_code, (rho_a, rho_b),
1,
relativity,
1,
verbose=verbose)[:2]
wva, wvb = numint._uks_gga_wv0((rho_a, rho_b), vxc, weight)
vtmp = numpy.zeros((3, nao, nao))
rks_grad._gga_grad_sum_(vtmp, mol, ao, wva, mask, ao_loc)
vmat[0] += vtmp
excsum += numpy.einsum('r,r,nxr->nx', exc, rho_a[0] + rho_b[0],
weight1)
excsum[atm_id] += numpy.einsum('xij,ji->x', vtmp, dms[0]) * 2
vtmp = numpy.zeros((3, nao, nao))
rks_grad._gga_grad_sum_(vtmp, mol, ao, wvb, mask, ao_loc)
vmat[1] += vtmp
excsum[atm_id] += numpy.einsum('xij,ji->x', vtmp, dms[1]) * 2
elif xctype == 'NLC':
raise NotImplementedError('NLC')
elif xctype == 'MGGA':
ao_deriv = 2
for atm_id, (coords, weight, weight1) \
in enumerate(rks_grad.grids_response_cc(grids)):
sh0, sh1 = aoslices[atm_id][:2]
mask = gen_grid.make_mask(mol, coords)
ao = ni.eval_ao(mol, coords, deriv=ao_deriv, non0tab=mask)
rho_a = make_rho(0, ao[:10], mask, xctype)
rho_b = make_rho(1, ao[:10], mask, xctype)
exc, vxc = ni.eval_xc(xc_code, (rho_a, rho_b),
1,
relativity,
1,
verbose=verbose)[:2]
wva, wvb = numint._uks_mgga_wv0((rho_a, rho_b), vxc, weight)
vtmp = numpy.zeros((3, nao, nao))
rks_grad._gga_grad_sum_(vtmp, mol, ao, wva, mask, ao_loc)
# *2 because wv[5] is scaled by 0.5 in _uks_mgga_wv0
rks_grad._tau_grad_dot_(vtmp, mol, ao, wva[5] * 2, mask, ao_loc,
True)
vmat[0] += vtmp
excsum += numpy.einsum('r,r,nxr->nx', exc, rho_a[0] + rho_b[0],
weight1)
excsum[atm_id] += numpy.einsum('xij,ji->x', vtmp, dms[0]) * 2
vtmp = numpy.zeros((3, nao, nao))
rks_grad._gga_grad_sum_(vtmp, mol, ao, wvb, mask, ao_loc)
rks_grad._tau_grad_dot_(vtmp, mol, ao, wvb[5] * 2, mask, ao_loc,
True)
vmat[1] += vtmp
excsum[atm_id] += numpy.einsum('xij,ji->x', vtmp, dms[1]) * 2
# - sign because nabla_X = -nabla_x
return excsum, -vmat